Hyperbolic sine: Integration (formula 01.19.21.2685)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function and trigonometric functions

Involving powers of the direct function and trigonometric functions

Involving powers of cos

Involving cos^mu(c z+d)sinh^v(a z)

http://functions.wolfram.com/01.19.21.2685.01

Input Form

Integrate[Cos[d + c z]^\[Mu] Sinh[a z]^v, z] == (I/2)^v Cos[d + c z]^\[Mu] (-((1/(c \[Mu])) ((I Binomial[v, v/2] Hypergeometric2F1[-(\[Mu]/2), -\[Mu], 1 - \[Mu]/2, -E^(2 I (d + c z))] (-1 + Mod[v, 2]))/ (1 + E^(2 I (d + c z)))^\[Mu])) + Sum[(-1)^k Binomial[v, k] ((E^(I Pi v + 2 a k z - a v z) Hypergeometric2F1[(I (2 a k - a v + I c \[Mu]))/(2 c), -\[Mu], (I (2 a k - a v + I c (-2 + \[Mu])))/(2 c), -E^(-2 I (d + c z))])/ (2 a k - a v + I c \[Mu]) - (E^(a (-2 k + v) z) Hypergeometric2F1[ (I (a (-2 k + v) + I c \[Mu]))/(2 c), -\[Mu], (I (a (-2 k + v) + I c (-2 + \[Mu])))/(2 c), -E^(-2 I (d + c z))])/ (2 a k - a v - I c \[Mu])), {k, 0, Floor[(1/2) (-1 + v)]}]/ (I^v (1 + E^(-2 I (d + c z)))^\[Mu])) /; Element[v, Integers] && v > 0

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