Hyperbolic cosine integral: Integration (formula 01.20.21.1991)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic, exponential and trigonometric functions

Involving rational functions of sinh, sin and exp

Involving e^{p z}sin(e z)cosh(d z)/a+b sinh²(c z)

http://functions.wolfram.com/01.20.21.1991.01

Input Form

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

Standard Form

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MathML Form

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

2002-12-18