Hyperbolic sine: Integration (formula 01.19.21.1135)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and a power functions

Involving powers of cos and power

Involving z^alpha-1 cos^m(b z^r+e) sinh(c z^r)

http://functions.wolfram.com/01.19.21.1135.01

Input Form

Integrate[z^n Cos[b z^2 + e]^m Sinh[c z^2], z] == (-1)^m 2^(-2 - m) z^(1 + n) Binomial[m, m/2] ((-((-c) z^2)^((1/2) (-1 - n))) Gamma[(1 + n)/2, (-c) z^2] + (c z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, c z^2]) (1 - Mod[m, 2]) - 2^(-2 - m) z^(1 + n) Sum[Binomial[m, k] (E^(-2 I e k + I e m) ((-c + 2 I b k - I b m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-c + 2 I b k - I b m) z^2] - E^(-2 I e k + I e m) ((c + 2 I b k - I b m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (c + 2 I b k - I b m) z^2] + E^(2 I e k - I e m) ((-c - 2 I b k + I b m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-c - 2 I b k + I b m) z^2] - E^(2 I e k - I e m) ((c - 2 I b k + I b m) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (c - 2 I b k + I b m) z^2]), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

Standard Form

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MathML Form

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type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> 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<cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> e </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> e </ci> <ci> k </ci> </apply> </apply> </apply> </apply> 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Rule Form

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

2002-12-18