Hyperbolic sine: Integration (formula 01.19.21.1246)

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/01.19.21.1246.01

Input Form

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