Hyperbolic cosine integral: Integration (formula 01.20.21.1879)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic and exponential functions

Involving rational functions of sinh and exp

Involving e^{p z}(a+b sinh²(d z))^-ncosh(c z)

http://functions.wolfram.com/01.20.21.1879.01

Input Form

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