Hyperbolic secant: Integration (formula 01.24.21.0328)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Integration

Indefinite integration

Involving functions of the direct function

Involving algebraic functions of the direct function

Involving ((a+b sech²(c z))ⁿ)^beta

http://functions.wolfram.com/01.24.21.0328.01

Input Form

Integrate[Sech[c z] Sqrt[(a + b Sech[c z]^2)^3], z] == (2 Cosh[c z]^3 Sqrt[(a + b Sech[c z]^2)^3] (2 I Sqrt[2] (2 a^2 + 3 a b + b^2) Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticE[I c z, a/(a + b)] - I Sqrt[2] (3 a^2 + 5 a b + 2 b^2) Sqrt[(a + 2 b + a Cosh[2 c z])/(a + b)] EllipticF[I c z, a/(a + b)] + Tanh[c z] (4 a^2 + 11 a b + 4 b^2 + 2 a (2 a + b) Cosh[2 c z] + b (a + 2 b) Sech[c z]^2 + a b Tanh[c z]^2)))/ (3 c (a + 2 b + a Cosh[2 c z])^2)

Standard Form

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MathML Form

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</apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> a </ci> </apply> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> <ci> a </ci> </apply> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <imaginaryi /> <apply> <ci> EllipticE </ci> <apply> <times /> <imaginaryi /> <ci> 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Rule Form

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

2002-12-18