Hyperbolic sine: Integration (formula 01.19.21.2232)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function and exponential function

Involving products of powers of two direct functions and exponential function

Involving e^{p z}sinh^m(c z) sinh^v(a z+b)

http://functions.wolfram.com/01.19.21.2232.01

Input Form

Integrate[E^(p z) Sinh[c z]^\[Mu] Sinh[b + a z]^v, z] == (1/(p - c \[Mu])) (((I/2)^v E^(p z) Binomial[v, v/2] Hypergeometric2F1[(p - c \[Mu])/(2 c), -\[Mu], (1/2) (2 + p/c - \[Mu]), E^(2 c z)] (1 - Mod[v, 2]) Sinh[c z]^\[Mu])/(1 - E^(2 c z))^\[Mu]) + (Sinh[c z]^\[Mu] Sum[(-1)^k E^(b (-2 k + v)) Binomial[v, k] ((E^(2 I ((Pi v)/2 + I b (-2 k + v)) + (p - a (-2 k + v)) z) Hypergeometric2F1[(p - a (-2 k + v) - c \[Mu])/(2 c), -\[Mu], (1/2) (2 + (p - a (-2 k + v))/c - \[Mu]), E^(2 c z)])/ (p - a (-2 k + v) - c \[Mu]) + (E^((p + a (-2 k + v)) z) Hypergeometric2F1[(p + a (-2 k + v) - c \[Mu])/(2 c), -\[Mu], (1/2) (2 + (p + a (-2 k + v))/c - \[Mu]), E^(2 c z)])/ (p + a (-2 k + v) - c \[Mu])), {k, 0, Floor[(1/2) (-1 + v)]}])/ (2^v (1 - E^(2 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