Hyperbolic cosine integral: Integration (formula 01.20.21.3396)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving functions of the direct function and trigonometric functions

Involving rational functions of the direct function and trigonometric functions

Involving sin

Involving sin(d z)cosh(e z)/a+b cosh²(c z)

http://functions.wolfram.com/01.20.21.3396.01

Input Form

Integrate[(Sin[d z] Cosh[e z])/(a + b Cosh[c z]^2), z] == (-(1/4)) I ((E^((2 c - I d - e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + ((-I) d - e)/(2 c), 1, 2 + ((-I) d - e)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + ((-I) d - e)/(2 c), 1, 2 + ((-I) d - e)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/ (Sqrt[a] b Sqrt[a + b] (2 c - I d - e)) - (E^((2 c + I d - e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I d - e)/(2 c), 1, 2 + (I d - e)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I d - e)/(2 c), 1, 2 + (I d - e)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/ (Sqrt[a] b Sqrt[a + b] (2 c + I d - e)) + (E^((2 c - I d + e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + ((-I) d + e)/(2 c), 1, 2 + ((-I) d + e)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + ((-I) d + e)/(2 c), 1, 2 + ((-I) d + e)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/ (Sqrt[a] b Sqrt[a + b] (2 c - I d + e)) - (E^((2 c + I d + e) z) ((-2 a - b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 + (I d + e)/(2 c), 1, 2 + (I d + e)/(2 c), -((b E^(2 c z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (2 a + b - 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 + (I d + e)/(2 c), 1, 2 + (I d + e)/(2 c), -((b E^(2 c z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))]))/ (Sqrt[a] b Sqrt[a + b] (2 c + I d + e)))

Standard Form

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MathML Form

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Rule Form

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

2002-12-18