Hyperbolic sine: Integration (formula 01.19.21.1752)

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 sinh(2c z))^beta sinh^v(c z)

http://functions.wolfram.com/01.19.21.1752.01

Input Form

Integrate[Sqrt[a + b Sinh[2 c z]] Sinh[c z]^2, z] == (-2 b^2 + 2 b^2 Cosh[4 c z] + 4 a b Sinh[2 c z] - 12 I (a - I b) b EllipticE[(1/4) (Pi - 4 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[2 c z])/(a - I b)] + 3 a Sqrt[(a + b Sinh[2 c z])/(1 + Cosh[c z])^2] Sqrt[Sech[(c z)/2]^4 (a + b Sinh[2 c z])] + 4 a Cosh[c z] Sqrt[(a + b Sinh[2 c z])/(1 + Cosh[c z])^2] Sqrt[Sech[(c z)/2]^4 (a + b Sinh[2 c z])] + a Cosh[2 c z] Sqrt[(a + b Sinh[2 c z])/(1 + Cosh[c z])^2] Sqrt[Sech[(c z)/2]^4 (a + b Sinh[2 c z])])/ (24 b c Sqrt[a + b Sinh[2 c z]])

Standard Form

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MathML Form

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</cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <imaginaryi /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> 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<cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <cosh /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <sech /> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <sinh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </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