Hyperbolic cosine integral: Integration (formula 01.20.21.4690)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving functions of the direct function, hyperbolic and exponential functions

Involving rational functions of the direct function, hyperbolic and exponential functions

Involving sinh and exp

Involving e^{p z}cosh(e z)sinh(d z)/a+b cosh²(c z)

http://functions.wolfram.com/01.20.21.4690.01

Input Form

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