Hyperbolic cosine integral: Integration (formula 01.20.21.1318)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving rational functions of cos and exp

Involving e^{p z}(a+b cos²(d z))^-ncosh(c z)

http://functions.wolfram.com/01.20.21.1318.01

Input Form

Integrate[(E^(p z) Cosh[c z])/(a + b Cos[d z]^2)^2, z] == (1/2) (-((E^((-c + 2 I d + p) z) ((-2 a - b) (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 - (I (-c + p))/(2 d), 1, 2 - (I (-c + p))/(2 d), -((b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] - (2 a + b) (-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 - (I (-c + p))/(2 d), 1, 2 - (I (-c + p))/(2 d), -((b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))] + 2 Sqrt[a] ((2 a^(3/2) + 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b]) Hypergeometric2F1[1 - (I (-c + p))/(2 d), 2, 2 - (I (-c + p))/(2 d), -((b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (-2 a^(3/2) - 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b]) Hypergeometric2F1[ 1 - (I (-c + p))/(2 d), 2, 2 - (I (-c + p))/(2 d), -((b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))])))/ (2 a^(3/2) b (a + b)^(3/2) (-c + 2 I d + p))) - (E^((c + 2 I d + p) z) ((-2 a - b) (2 a + b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[1 - (I (c + p))/(2 d), 1, 2 - (I (c + p))/(2 d), -((b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] - (2 a + b) (-2 a - b + 2 Sqrt[a] Sqrt[a + b]) Hypergeometric2F1[ 1 - (I (c + p))/(2 d), 1, 2 - (I (c + p))/(2 d), -((b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))] + 2 Sqrt[a] ((2 a^(3/2) + 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b]) Hypergeometric2F1[1 - (I (c + p))/(2 d), 2, 2 - (I (c + p))/(2 d), -((b E^(2 I d z))/(2 a + b - 2 Sqrt[a] Sqrt[a + b]))] + (-2 a^(3/2) - 2 Sqrt[a] b + 2 a Sqrt[a + b] + b Sqrt[a + b]) Hypergeometric2F1[1 - (I (c + p))/(2 d), 2, 2 - (I (c + p))/(2 d), -((b E^(2 I d z))/(2 a + b + 2 Sqrt[a] Sqrt[a + b]))])))/ (2 a^(3/2) b (a + b)^(3/2) (c + 2 I d + p)))

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