Hyperbolic cosine integral: Integration (formula 01.20.21.0788)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric functions

Involving powers of cos

Involving cos^m(b z^r+e) cosh(f z+g)

http://functions.wolfram.com/01.20.21.0788.01

Input Form

Integrate[Cos[b z^2 + e]^m Cosh[f z + g], z] == ((-(1/2))^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[g + f z])/f + 2^(-(1/2) - m) Sqrt[Pi] Sum[Binomial[m, s] ((1/Sqrt[b m - 2 b s]) (Cosh[g - I e (-m + 2 s) + (I f^2)/(4 (b m - 2 b s))] FresnelC[((-I) f + 2 (b m - 2 b s) z)/ (Sqrt[2 Pi] Sqrt[b m - 2 b s])] + I FresnelS[((-I) f + 2 (b m - 2 b s) z)/(Sqrt[2 Pi] Sqrt[b m - 2 b s])] Sinh[g - I e (-m + 2 s) + (I f^2)/(4 (b m - 2 b s))]) + (1/Sqrt[(-b) m + 2 b s]) (Cosh[g + I e (-m + 2 s) + (I f^2)/(4 ((-b) m + 2 b s))] FresnelC[((-I) f + 2 ((-b) m + 2 b s) z)/(Sqrt[2 Pi] Sqrt[(-b) m + 2 b s])] + I FresnelS[((-I) f + 2 ((-b) m + 2 b s) z)/(Sqrt[2 Pi] Sqrt[(-b) m + 2 b s])] Sinh[g + I e (-m + 2 s) + (I f^2)/(4 ((-b) m + 2 b s))])), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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<cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> s </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> f </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus 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<apply> <times /> <ci> b </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> s </ci> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> m </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <ci> s </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> f </ci> </apply> <apply> <times /> <cn type='integer'> 2 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Rule Form

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

2002-12-18