Hyperbolic cosine integral: Integration (formula 01.20.21.1288)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving powers of cos and exp

Involving e^{p z}cos^mu(c z+d) cosh(a z+b)

http://functions.wolfram.com/01.20.21.1288.01

Input Form

Integrate[E^(p z) Cos[d + c z]^m Cosh[b + a z], z] == (Binomial[m, m/2] (1 - Mod[m, 2]) ((E^(p z) (p Cosh[b + a z] - a Sinh[b + a z]))/(p^2 - a^2)))/2^m + 2^(1 - m) Sum[Binomial[m, k] ((E^(p z) (Cosh[b + a z] (p (-a^2 + c^2 (-2 k + m)^2 + p^2) Cos[(2 k - m) (d + c z)] + c (2 k - m) (a^2 + c^2 (-2 k + m)^2 + p^2) Sin[(2 k - m) (d + c z)]) + a ((a^2 + c^2 (-2 k + m)^2 - p^2) Cos[(2 k - m) (d + c z)] + 2 c (-2 k + m) p Sin[(2 k - m) (d + c z)]) Sinh[b + a z]))/ (a^4 + 2 a^2 (c^2 (-2 k + m)^2 - p^2) + (c^2 (-2 k + m)^2 + p^2)^2)), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> m </ci> <ci> k </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <ci> p </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <cosh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> p </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> c </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <ci> p </ci> <apply> <sin /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2002-12-18