Hyperbolic cosecant: Integration (formula 01.23.21.0206)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic and trigonometric functions

Involving powers of cos and powers of sinh

Involving cos^m(a z) sinh^u(b z) csch(c z)

http://functions.wolfram.com/01.23.21.0206.01

Input Form

Integrate[Cos[a z]^m Sinh[c z]^\[Mu] Csch[c z], z] == ((1/(c (\[Mu] - 1))) 2^(1 - m - \[Mu]) E^(c z) (-E^((-c) z) + E^(c z))^ \[Mu] Binomial[m, m/2] Hypergeometric2F1[(1 - \[Mu])/2, -\[Mu] + 1, (3 - \[Mu])/2, E^(2 c z)] (1 - Mod[m, 2]))/(1 - E^(2 c z))^\[Mu] - (2^(1 - m - \[Mu]) E^(c z) (-E^((-c) z) + E^(c z))^\[Mu] Sum[Binomial[m, k] ((E^(I a (2 k - m) z) Hypergeometric2F1[ (I a (2 k - m) - c \[Mu] + c)/(2 c), -\[Mu] + 1, (I a (2 k - m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/ (I a (2 k - m) + c (-\[Mu] + 1)) + (E^(I a (-2 k + m) z) Hypergeometric2F1[(I a (-2 k + m) - c \[Mu] + c)/(2 c), -\[Mu] + 1, (I a (-2 k + m) + c (3 - \[Mu]))/(2 c), E^(2 c z)])/ (I a (-2 k + m) + c (-\[Mu] + 1))), {k, 0, Floor[(1/2) (-1 + m)]}])/ (1 - E^(2 c z))^\[Mu] /; Element[m, Integers] && m > 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