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 Coth

 http://functions.wolfram.com/01.22.21.0214.01

 Input Form

 Integrate[z^n Sin[a z] Sinh[b z] Coth[c z], z] == (-(1/4)) I n! (E^((I a - b) z) Sum[(1/(-j + n)!) ((-1)^j (I a - b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[v, 1], \[Ellipsis], Subscript[v, 1 + j], 1}, {1 + Subscript[v, 1], \[Ellipsis], 1 + Subscript[v, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(((-I) a + b) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a + b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[w, 1], \[Ellipsis], Subscript[w, 1 + j], 1}, {1 + Subscript[w, 1], \[Ellipsis], 1 + Subscript[w, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^((I a - b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j (I a - b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[x, 1], \[Ellipsis], Subscript[x, 1 + j], 1}, {1 + Subscript[x, 1], \[Ellipsis], 1 + Subscript[x, 1 + j]}, E^(2 c z)]), {j, 0, n}] + E^(((-I) a + b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a + b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[ {Subscript[y, 1], \[Ellipsis], Subscript[y, 1 + j], 1}, {1 + Subscript[y, 1], \[Ellipsis], 1 + Subscript[y, 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^(((-I) a - b) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a - b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[z, 1], \[Ellipsis], Subscript[z, 1 + j], 1}, {1 + Subscript[z, 1], \[Ellipsis], 1 + Subscript[z, 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^((I a + b) z) Sum[(1/(-j + n)!) ((-1)^j (I a + b)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], 1 + j], 1}, {1 + Subscript[\[Alpha], 1], \[Ellipsis], 1 + Subscript[\[Alpha], 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^(((-I) a - b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j ((-I) a - b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], 1 + j], 1}, {1 + Subscript[\[Beta], 1], \[Ellipsis], 1 + Subscript[\[Beta], 1 + j]}, E^(2 c z)]), {j, 0, n}] - E^((I a + b + 2 c) z) Sum[(1/(-j + n)!) ((-1)^j (I a + b + 2 c)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[\[Gamma], 1], \[Ellipsis], Subscript[\[Gamma], 1 + j], 1}, {1 + Subscript[\[Gamma], 1], \[Ellipsis], 1 + Subscript[\[Gamma], 1 + j]}, E^(2 c z)]), {j, 0, n}]) /; Subscript[v, 1] == Subscript[v, 2] == \[Ellipsis] == Subscript[v, n + 1] == (I a - b)/(2 c) && Subscript[w, 1] == Subscript[w, 2] == \[Ellipsis] == Subscript[w, n + 1] == ((-I) a + b)/(2 c) && Subscript[x, 1] == Subscript[x, 2] == \[Ellipsis] == Subscript[x, n + 1] == (I a - b + 2 c)/(2 c) && Subscript[y, 1] == Subscript[y, 2] == \[Ellipsis] == Subscript[y, n + 1] == ((-I) a + b + 2 c)/(2 c) && Subscript[z, 1] == Subscript[z, 2] == \[Ellipsis] == Subscript[z, n + 1] == ((-I) a - b)/(2 c) && Subscript[\[Alpha], 1] == Subscript[\[Alpha], 2] == \[Ellipsis] == Subscript[\[Alpha], n + 1] == (I a + b)/(2 c) && Subscript[\[Beta], 1] == Subscript[\[Beta], 2] == \[Ellipsis] == Subscript[\[Beta], n + 1] == ((-I) a - b + 2 c)/(2 c) && Subscript[\[Gamma], 1] == Subscript[\[Gamma], 2] == \[Ellipsis] == Subscript[\[Gamma], n + 1] == (I a + b + 2 c)/(2 c) && Element[n, Integers] && n >= 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