Hyperbolic sine: Integration (formula 01.19.21.1815)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function

Involving algebraic functions of the direct function

Involving (a+b sinh²(c z))^beta sinh^nu(c z)

http://functions.wolfram.com/01.19.21.1815.01

Input Form

Integrate[Sinh[c z]^4/(a + b Sinh[c z]^2)^(5/2), z] == (2 I a^2 (a - 2 b) ((2 a - b + b Cosh[2 c z])/a)^(3/2) EllipticE[I c z, b/a] - I a (2 a^2 - 5 a b + 3 b^2) ((2 a - b + b Cosh[2 c z])/a)^(3/2) EllipticF[I c z, b/a] - Sqrt[2] b (-a^2 + 4 a b - 2 b^2 - (a - 2 b) b Cosh[2 c z]) Sinh[2 c z])/ (3 (a - b)^2 b^2 c (2 a - b + b Cosh[2 c z])^(3/2))

Standard Form

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MathML Form

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</apply> <cn type='rational'> 5 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> <imaginaryi /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <ci> b </ci> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </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