Hyperbolic sine: Integration (formula 01.19.21.1245)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving trigonometric and exponential functions

Involving rational functions of sin and exp

Involving e^{p z}(a+b sin²(d z))^-nsinh(c z)

http://functions.wolfram.com/01.19.21.1245.01

Input Form

Integrate[(E^(p z) Sinh[c z])/(a + b Sin[d z]^2)^2, z] == -(-((1/(-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])])))) + (1/(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])]))))/ (4 a^(3/2) b (a + b)^(3/2))

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