Hyperbolic sine: Integration (formula 01.19.21.3255)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function, trigonometric and exponential functions

Involving powers of the direct function, trigonometric and exponential functions

Involving cot and exp

http://functions.wolfram.com/01.19.21.3255.01

Input Form

Integrate[E^(p z) Cot[b z] Sinh[c z]^v, z] == (-(1/((2 b - I p) p))) (I/2)^v Binomial[v, v/2] (1 - Mod[v, 2]) (E^(p z) (2 I b + p) Hypergeometric2F1[-((I p)/(2 b)), 1, 1 - (I p)/(2 b), E^(2 I b z)] + E^((2 I b + p) z) p Hypergeometric2F1[1 - (I p)/(2 b), 1, 2 - (I p)/(2 b), E^(2 I b z)]) + (I Sum[(-1)^k Binomial[v, k] ((-(1/(p + c (-2 k + v)))) E^((p + c (-2 k + v)) z) Hypergeometric2F1[(I (2 c k - p - c v))/ (2 b), 1, 1 + (I (2 c k - p - c v))/(2 b), E^(2 I b z)] - E^((2 I b + p + c (-2 k + v)) z) (Hypergeometric2F1[ (2 b + I (2 c k - p - c v))/(2 b), 1, (4 b + I (2 c k - p - c v))/ (2 b), E^(2 I b z)]/(2 I b + p + c (-2 k + v))) + (-1)^v ((-(1/(2 c k + p - c v))) E^((2 c k + p - c v) z) Hypergeometric2F1[-((I (2 c k + p - c v))/(2 b)), 1, 1 - (I (2 c k + p - c v))/(2 b), E^(2 I b z)] - E^((2 I b + 2 c k + p - c v) z) (Hypergeometric2F1[ (2 b - I (2 c k + p - c v))/(2 b), 1, (4 b - I (2 c k + p - c v))/(2 b), E^(2 I b z)]/ (2 I b + 2 c k + p - c v)))), {k, 0, Floor[(1/2) (-1 + v)]}])/ 2^v /; Element[v, Integers] && v > 0

Standard Form

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MathML Form

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<cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> v </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <plus /> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> v </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> p </ci> <apply> <times /> <ci> c </ci> <apply> <plus /> <ci> v </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> 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Rule Form

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

2002-12-18