Hyperbolic sine: Integration (formula 01.19.21.3007)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Integration

Indefinite integration

Involving functions of the direct function, trigonometric and a power functions

Involving powers of the direct function, trigonometric and a power functions

Involving cos and power

Involving z^alpha-1 cos(b z^r+e) sinh^v(c z^r+g)

http://functions.wolfram.com/01.19.21.3007.01

Input Form

Integrate[z^(\[Alpha] - 1) Cos[b z^r + e] Sinh[c z^r + g]^v, z] == (-(1/r)) ((2^(-1 - v) z^\[Alpha] Binomial[v, v/2] ((E^(I e) Gamma[\[Alpha]/r, (-I) b z^r])/((-I) b z^r)^(\[Alpha]/r) + Gamma[\[Alpha]/r, I b z^r]/(E^(I e) (I b z^r)^(\[Alpha]/r))) (1 - Mod[v, 2]))/I^v + 2^(-1 - v) z^\[Alpha] Sum[(-1)^s Binomial[v, s] ((E^(I e - 2 g s + g v) Gamma[\[Alpha]/r, ((-I) b + 2 c s - c v) z^r])/(((-I) b + 2 c s - c v) z^r)^ (\[Alpha]/r) + (E^((-I) e - 2 g s + g v) Gamma[\[Alpha]/r, (I b + 2 c s - c v) z^r])/((I b + 2 c s - c v) z^r)^(\[Alpha]/r) + ((-1)^v E^(I e + 2 g s - g v) Gamma[\[Alpha]/r, ((-I) b - 2 c s + c v) z^r])/(((-I) b - 2 c s + c v) z^r)^ (\[Alpha]/r) + ((-1)^v E^((-I) e + 2 g s - g v) Gamma[\[Alpha]/r, (I b - 2 c s + c v) z^r])/ ((I b - 2 c s + c v) z^r)^(\[Alpha]/r)), {s, 0, Floor[(1/2) (-1 + v)]}]) /; 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