Hyperbolic cosine integral: Integration (formula 01.20.21.1232)

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 sin and exp

Involving e^{b z^r+e} sin^m(a z^r+q) cosh(c z^r+g)

http://functions.wolfram.com/01.20.21.1232.01

Input Form

Integrate[E^(b Sqrt[z] + e) Sin[a Sqrt[z] + q]^m Cosh[c Sqrt[z] + g], z] == (((E^(e - g + (b - c) Sqrt[z]) (-1 + b Sqrt[z] - c Sqrt[z]))/(-b + c)^2 + (E^(e + g + (b + c) Sqrt[z]) (-1 + b Sqrt[z] + c Sqrt[z]))/(b + c)^2) Binomial[m, m/2] (1 - Mod[m, 2]))/2^m + Sum[(-1)^k ((E^(e - g - (I m Pi)/2 + I (-2 k + m) q + (b - c + I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] - (c - I a (-2 k + m)) Sqrt[z]))/(-b + c - I a (-2 k + m))^2 + (E^(e + g + (I m Pi)/2 - I (-2 k + m) q + (b + c - I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] + (c - I a (-2 k + m)) Sqrt[z]))/ (b + c - I a (-2 k + m))^2 + (E^(e - g + (I m Pi)/2 - I (-2 k + m) q + (b - c - I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] - (c + I a (-2 k + m)) Sqrt[z]))/ (-b + c + I a (-2 k + m))^2 + (E^(e + g - (I m Pi)/2 + I (-2 k + m) q + (b + c + I a (-2 k + m)) Sqrt[z]) (-1 + b Sqrt[z] + (c + I a (-2 k + m)) Sqrt[z]))/ (b + c + I a (-2 k + m))^2) Binomial[m, k], {k, 0, Floor[(1/2) (-1 + m)]}]/2^m /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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<mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> - </mo> <mi> g </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> m </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> 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Rule Form

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

2002-12-18