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 Coth

 http://functions.wolfram.com/01.22.21.0228.01

 Input Form

 Integrate[z^n E^(p z) Sin[a z] Cosh[b z] Coth[c z], z] == (I/4) n! ((-E^(((-I) a - b + p) z)) Sum[(1/(-j + n)!) (-1)^j ((-I) a - b + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[ {Subscript[q, 1], \[Ellipsis], Subscript[q, 1 + j], 1}, {1 + Subscript[q, 1], \[Ellipsis], 1 + Subscript[q, 1 + j]}, E^(2 c z)], {j, 0, n}] - E^(((-I) a - b + 2 c + p) z) Sum[(1/(-j + n)!) (-1)^j ((-I) a - b + 2 c + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[r, 1], \[Ellipsis], Subscript[r, 1 + j], 1}, {1 + Subscript[r, 1], \[Ellipsis], 1 + Subscript[r, 1 + j]}, E^(2 c z)], {j, 0, n}] + E^((I a - b + 2 c + p) z) Sum[(1/(-j + n)!) (-1)^j (I a - b + 2 c + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[u, 1], \[Ellipsis], Subscript[u, 1 + j], 1}, {1 + Subscript[u, 1], \[Ellipsis], 1 + Subscript[u, 1 + j]}, E^(2 c z)], {j, 0, n}] + E^((I a - b + p) z) Sum[(1/(-j + n)!) (-1)^j (I a - b + p)^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[t, 1], \[Ellipsis], Subscript[t, 1 + j], 1}, {1 + Subscript[t, 1], \[Ellipsis], 1 + Subscript[t, 1 + j]}, E^(2 c z)], {j, 0, n}] - E^(((-I) a + b + p) z) Sum[(1/(-j + n)!) (-1)^j ((-I) a + b + p)^(-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 + p) z) Sum[(1/(-j + n)!) (-1)^j ((-I) a + b + 2 c + p)^(-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 + p) z) Sum[(1/(-j + n)!) (-1)^j (I a + b + p)^(-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 + 2 c + p) z) Sum[(1/(-j + n)!) (-1)^j (I a + b + 2 c + p)^(-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}]) /; Subscript[q, 1] == Subscript[q, 2] == \[Ellipsis] == Subscript[q, n + 1] == ((-I) a + p - b)/(2 c) && Subscript[r, 1] == Subscript[r, 2] == \[Ellipsis] == Subscript[r, n + 1] == ((-I) a + p - b + 2 c)/(2 c) && Subscript[t, 1] == Subscript[t, 2] == \[Ellipsis] == Subscript[t, n + 1] == (I a + p - b)/(2 c) && Subscript[u, 1] == Subscript[u, 2] == \[Ellipsis] == Subscript[u, n + 1] == (I a + p - b + 2 c)/(2 c) && Subscript[w, 1] == Subscript[w, 2] == \[Ellipsis] == Subscript[w, n + 1] == ((-I) a + p + b)/(2 c) && Subscript[x, 1] == Subscript[x, 2] == \[Ellipsis] == Subscript[x, n + 1] == ((-I) a + p + b + 2 c)/(2 c) && Subscript[z, 1] == Subscript[z, 2] == \[Ellipsis] == Subscript[z, n + 1] == (I a + p + b)/(2 c) && Subscript[\[Alpha], 1] == Subscript[\[Alpha], 2] == \[Ellipsis] == Subscript[\[Alpha], n + 1] == (I a + p + b + 2 c)/(2 c) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[u, Integers] && u > 0 && Element[v, Integers] && v > 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