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 Coth

 http://functions.wolfram.com/01.22.21.0061.01

 Input Form

 Integrate[z^n E^(p z) Sin[b z]^m Coth[c z], z] == (-2^(-m)) Binomial[m, m/2] n! (1 - Mod[m, 2]) (E^(p z) Sum[(1/(-j + n)!) ((-1)^j p^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[b, 1], \[Ellipsis], Subscript[b, 1 + j], 1}, {1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + p) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[c, 1], \[Ellipsis], Subscript[c, 1 + j], 1}, {1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, 1 + j]}, E^(2 c z)]), {j, 0, n}]) - (n! Sum[(-1)^k Binomial[m, k] ((-1)^m (E^(((-I) b (-2 k + m) + p) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) b (-2 k + m) + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[f, 1], \[Ellipsis], Subscript[f, 1 + j], 1}, {1 + Subscript[f, 1], \[Ellipsis], 1 + Subscript[f, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c - I b (-2 k + m) + p) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I b (-2 k + m) + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[g, 1], \[Ellipsis], Subscript[g, 1 + j], 1}, {1 + Subscript[g, 1], \[Ellipsis], 1 + Subscript[g, 1 + j]}, E^(2 c z)]), {j, 0, n}]) + E^((I b (-2 k + m) + p) z) Sum[(1/(-j + n)!) ((-1)^j (I b (-2 k + m) + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[h, 1], \[Ellipsis], Subscript[h, 1 + j], 1}, {1 + Subscript[h, 1], \[Ellipsis], 1 + Subscript[h, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((2 c + I b (-2 k + m) + p) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I b (-2 k + m) + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[i, 1], \[Ellipsis], Subscript[i, 1 + j], 1}, {1 + Subscript[i, 1], \[Ellipsis], 1 + Subscript[i, 1 + j]}, E^(2 c z)]), {j, 0, n}]), {k, 0, Floor[(1/2) (-1 + m)]}])/(2 I)^m /; Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] == p/(2 c) && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == (p + 2 c)/(2 c) && Subscript[f, 1] == Subscript[f, 2] == \[Ellipsis] == Subscript[f, n + 1] == (p - I b (-2 k + m))/(2 c) && Subscript[g, 1] == Subscript[g, 2] == \[Ellipsis] == Subscript[g, n + 1] == (p - I b (-2 k + m) + 2 c)/(2 c) && Subscript[h, 1] == Subscript[h, 2] == \[Ellipsis] == Subscript[h, n + 1] == (p + I b (-2 k + m))/(2 c) && Subscript[i, 1] == Subscript[i, 2] == \[Ellipsis] == Subscript[i, n + 1] == (p + I b (-2 k + m) + 2 c)/(2 c) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0

 Standard Form

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 MathML Form

 Rule Form

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

 2002-12-18