Hyperbolic tangent: Integration (formula 01.21.21.0364)

Tanh

$Mathematica Notation$

Elementary Functions

Tanh[z]

Integration

Indefinite integration

Involving functions of the direct function and hyperbolic functions

Involving algebraic functions of the direct function and hyperbolic functions

Involving cosh

Involving cosh(c z)(a+b tanh²(c z))^beta

http://functions.wolfram.com/01.21.21.0364.01

Input Form

Integrate[Cosh[c z] Sqrt[a + b Tanh[c z]^2], z] == (I Sqrt[2] Sqrt[-(1/a)] a Cosh[c z] Coth[c z] (EllipticE[I ArcSinh[(Sqrt[-(1/a)] Sqrt[a - b + (a + b) Cosh[2 c z]])/ Sqrt[2]], -(a/b)] - EllipticF[ I ArcSinh[(Sqrt[-(1/a)] Sqrt[a - b + (a + b) Cosh[2 c z]])/Sqrt[2]], -(a/b)]) Sqrt[-(((a + b) Sinh[c z]^2)/a)] Sqrt[a + b Tanh[c z]^2])/ (c Sqrt[((a + b) Cosh[c z]^2)/b] Sqrt[a - b + (a + b) Cosh[2 c z]])

Standard Form

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MathML Form

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

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

2002-12-18