Hyperbolic secant: Integration (formula 01.24.21.0177)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic and a power functions

Involving powers of coth and power

Involving zⁿ coth^u(c z)

http://functions.wolfram.com/01.24.21.0177.01

Input Form

Integrate[z^n Coth[c z]^u Sech[c z], z] == 2 (-1)^u Binomial[u - 1, (u - 1)/2] n! (1 - Mod[u - 1, 2]) E^(c u z) Sum[(1/(-j + n)!) (-1)^j (c u)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 1 + j], u}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, E^(2 c z)], {j, 0, n}] + 2 (-1)^u E^(c u z) n! Sum[Binomial[u - 1, k] (E^(c (-2 k + u - 1) z) Sum[(1/(-j + n)!) (-1)^j (c (-2 k + 2 u - 1))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], u}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 c z)], {j, 0, n}] + Sum[(1/(-j + n)!) (-1)^j (c (2 k + 1))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], u}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, E^(2 c z)], {j, 0, n}]/ E^(c (-2 k + u - 1) z)), {k, 0, Floor[(1/2) (-2 + u)]}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == u/2 && Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == (-2 k + 2 u - 1)/2 && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (2 k + 1)/2 && Element[n, Integers] && n >= 0 && Element[u, Integers] && u > 0

Standard Form

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MathML Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18