Hyperbolic cosine integral: Integration (formula 01.20.21.3823)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving functions of the direct function, trigonometric and exponential functions

Involving powers of the direct function, trigonometric and exponential functions

Involving cos and rational functions of exp

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

http://functions.wolfram.com/01.20.21.3823.01

Input Form

Integrate[(E^(p z) Cos[e z] Cosh[c z]^v)/(a + b E^(d z))^n, z] == (2^(-1 - v) Binomial[v, v/2] ((-E^(((-I) e + p) z)) (I e + p) Hypergeometric2F1[((-I) e + p)/d, n, (d - I e + p)/d, -((b E^(d z))/a)] + E^((I e + p) z) (I e - p) Hypergeometric2F1[(I e + p)/d, n, (d + I e + p)/d, -((b E^(d z))/a)]) (1 - Mod[v, 2]))/a^n/((I e - p) (I e + p)) + (2^(-1 - v) Sum[Binomial[v, s] (((-E^((I e + p - c (2 s - v)) z)) ((-I) e + p + c (2 s - v)) Hypergeometric2F1[(I e + p - c (2 s - v))/d, n, (d + I e + p - c (2 s - v))/d, -((b E^(d z))/a)] + E^(((-I) e + p + c (2 s - v)) z) ((-I) e - p + c (2 s - v)) Hypergeometric2F1[((-I) e + p + c (2 s - v))/d, n, (d - I e + p + c (2 s - v))/d, -((b E^(d z))/a)])/ (((-I) e - p + c (2 s - v)) ((-I) e + p + c (2 s - v))) + (E^(((-I) e + p - c (2 s - v)) z) ((-I) e - p - c (2 s - v)) Hypergeometric2F1[((-I) e + p - c (2 s - v))/d, n, (d - I e + p - c (2 s - v))/d, -((b E^(d z))/a)] - E^((I e + p + c (2 s - v)) z) ((-I) e + p - c (2 s - v)) Hypergeometric2F1[(I e + p + c (2 s - v))/d, n, (d + I e + p + c (2 s - v))/d, -((b E^(d z))/a)])/ (((-I) e - p - c (2 s - v)) ((-I) e + p - c (2 s - v)))), {s, 0, Floor[(1/2) (-1 + v)]}])/a^n /; Element[n, Integers] && n > 0 && Element[v, Integers] && v > 0

Standard Form

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MathML Form

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

2002-12-18