Hyperbolic sine: Integration (formula 01.19.21.1333)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving rational functions of sin, cos and exp

Involving e^{p z}sinh(d z)(a sin²(e z)+b cos²(e z))^-n

http://functions.wolfram.com/01.19.21.1333.01

Input Form

Integrate[(E^(p z) Sinh[d z])/(a Sin[e z]^2 + b Cos[e z]^2)^2, z] == (I ((1/(-d + 2 I e + p)) (E^((-d + 2 I e + p) z) ((-(Sqrt[-a] + I Sqrt[b])^2) (a + b) Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 1, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 (a + b) Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 1, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] + I Sqrt[b])^2] - 2 I Sqrt[-a] Sqrt[b] ((Sqrt[-a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 + (I (d - p))/(2 e), 2, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 Hypergeometric2F1[1 + (I (d - p))/(2 e), 2, 2 + (I (d - p))/(2 e), ((-a + b) E^(2 I e z))/ (Sqrt[-a] + I Sqrt[b])^2]))) - (1/(d + 2 I e + p)) (E^((d + 2 I e + p) z) ((-(Sqrt[-a] + I Sqrt[b])^2) (a + b) Hypergeometric2F1[1 - (I (d + p))/(2 e), 1, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 (a + b) Hypergeometric2F1[ 1 - (I (d + p))/(2 e), 1, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] + I Sqrt[b])^2] - 2 I Sqrt[-a] Sqrt[b] ((Sqrt[-a] + I Sqrt[b])^2 Hypergeometric2F1[ 1 - (I (d + p))/(2 e), 2, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/(Sqrt[-a] - I Sqrt[b])^2] + (Sqrt[-a] - I Sqrt[b])^2 Hypergeometric2F1[1 - (I (d + p))/(2 e), 2, 2 - (I (d + p))/(2 e), ((-a + b) E^(2 I e z))/ (Sqrt[-a] + I Sqrt[b])^2])))))/(4 (-a)^(3/2) b^(3/2) (-a + b))

Standard Form

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</apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> 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<apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> d </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> e </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> b </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18