Hyperbolic cosine integral: Integration (formula 01.20.21.0779)

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) cosh(c z)

http://functions.wolfram.com/01.20.21.0779.01

Input Form

Integrate[Cos[b Sqrt[z]]^m Cosh[c z], z] == ((-(1/2))^m Binomial[m, m/2] (1 - Mod[m, 2]) Sinh[c z])/c + 2^(-1 - m) Sum[Binomial[m, s] ((1/((-I) c)^(3/2)) ((-b) Sqrt[2 Pi] (m - 2 s) (Cosh[(b m - 2 b s)^2/(4 c)] FresnelC[(b (m - 2 s) - 2 I c Sqrt[z])/(Sqrt[(-I) c] Sqrt[ 2 Pi])] + I FresnelS[(b (m - 2 s) - 2 I c Sqrt[z])/ (Sqrt[(-I) c] Sqrt[2 Pi])] Sinh[(b m - 2 b s)^2/(4 c)]) + 2 Sqrt[(-I) c] Sin[(b m - 2 b s) Sqrt[z] - I c z]) + (1/((-I) c)^(3/2)) (b Sqrt[2 Pi] (m - 2 s) (Cosh[(b m - 2 b s)^2/(4 c)] FresnelC[((-b) (m - 2 s) - 2 I c Sqrt[z])/(Sqrt[(-I) c] Sqrt[2 Pi])] + I FresnelS[((-b) (m - 2 s) - 2 I c Sqrt[z])/(Sqrt[(-I) c] Sqrt[ 2 Pi])] Sinh[(b m - 2 b s)^2/(4 c)]) - 2 Sqrt[(-I) c] Sin[(-((-b) m + 2 b s)) Sqrt[z] + I c z])), {s, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

Standard Form

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MathML Form

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<mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> 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<mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> 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Date Added to functions.wolfram.com (modification date)

2002-12-18