html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Tanh

 http://functions.wolfram.com/01.21.21.0195.01

 Input Form

 Integrate[z^n Cos[a z]^m Sinh[b z]^u Tanh[c z], z] == I^u 2^(-m - u) Binomial[m, m/2] Binomial[u, u/2] n! (1 - Mod[m, 2]) (1 - Mod[u, 2]) (-(z^(1 + n)/(1 + n)!) + 2 E^(2 c z) Sum[(1/(-j + n)!) ((-1)^j 2^(-1 - j) c^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, 2 + j]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, 1 + j]}, -E^(2 c z)]), {j, 0, n}]) + I^u 2^(-m - u) Binomial[u, u/2] n! (1 - Mod[u, 2]) Sum[Binomial[m, k] ((-E^((-I) a (-2 k + m) z)) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m))^(-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}] + E^((2 c - I a (-2 k + m)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[d, 1], \[Ellipsis], Subscript[d, 1 + j], 1}, {1 + Subscript[d, 1], \[Ellipsis], 1 + Subscript[d, 1 + j]}, -E^(2 c z)]), {j, 0, n}] - E^(I a (-2 k + m) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[e, 1], \[Ellipsis], Subscript[e, 1 + j], 1}, {1 + Subscript[e, 1], \[Ellipsis], 1 + Subscript[e, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((2 c + I a (-2 k + m)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m))^(-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}]), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-m - u) Binomial[m, m/2] n! (1 - Mod[m, 2]) Sum[(-1)^k Binomial[u, k] ((-E^(b (-2 k + u) z)) Sum[(1/(-j + n)!) ((-1)^j (b (-2 k + u))^ (-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^((2 c + b (-2 k + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + b (-2 k + u))^(-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}] + (-1)^u ((-E^((-b) (-2 k + u) z)) Sum[(1/(-j + n)!) ((-1)^j ((-b) (-2 k + u))^(-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}] + E^((2 c - b (-2 k + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - b (-2 k + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[k, 1], \[Ellipsis], Subscript[k, 1 + j], 1}, {1 + Subscript[k, 1], \[Ellipsis], 1 + Subscript[k, 1 + j]}, -E^(2 c z)]), {j, 0, n}])), {k, 0, Floor[(1/2) (-1 + u)]}] + 2^(-m - u) n! Sum[Binomial[m, k] Sum[(-1)^i Binomial[u, i] ((-E^(((-I) a (-2 k + m) + b (-2 i + u)) z)) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[l, 1], \[Ellipsis], Subscript[l, 1 + j], 1}, {1 + Subscript[l, 1], \[Ellipsis], 1 + Subscript[l, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((2 c - I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[m, 1], \[Ellipsis], Subscript[m, 1 + j], 1}, {1 + Subscript[m, 1], \[Ellipsis], 1 + Subscript[m, 1 + j]}, -E^(2 c z)]), {j, 0, n}] - E^((I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) + b (-2 i + u))^(-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[n, 1], \[Ellipsis], Subscript[n, 1 + j], 1}, {1 + Subscript[n, 1], \[Ellipsis], 1 + Subscript[n, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((2 c + I a (-2 k + m) + b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m) + b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[o, 1], \[Ellipsis], Subscript[o, 1 + j], 1}, {1 + Subscript[o, 1], \[Ellipsis], 1 + Subscript[o, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + (-1)^u ((-E^(((-I) a (-2 k + m) - b (-2 i + u)) z)) Sum[(1/(-j + n)!) ((-1)^j ((-I) a (-2 k + m) - b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[p, 1], \[Ellipsis], Subscript[p, 1 + j], 1}, {1 + Subscript[p, 1], \[Ellipsis], 1 + Subscript[p, 1 + j]}, -E^(2 c z)]), {j, 0, n}] + E^((2 c - I a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c - I a (-2 k + m) - b (-2 i + u))^ (-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 (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (I a (-2 k + m) - b (-2 i + u))^ (-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^((2 c + I a (-2 k + m) - b (-2 i + u)) z) Sum[(1/(-j + n)!) ((-1)^j (2 c + I a (-2 k + m) - b (-2 i + u))^ (-1 - j) z^(-j + n) HypergeometricPFQ[{Subscript[s, 1], \[Ellipsis], Subscript[s, 1 + j], 1}, {1 + Subscript[s, 1], \[Ellipsis], 1 + Subscript[s, 1 + j]}, -E^(2 c z)]), {j, 0, n}])), {i, 0, Floor[(1/2) (-1 + u)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 2] == 1 && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] == ((-I) a (-2 k + m))/(2 c) && Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] == (2 c - I a (-2 k + m))/(2 c) && Subscript[e, 1] == Subscript[e, 2] == \[Ellipsis] == Subscript[e, n + 1] == (I a (-2 k + m))/(2 c) && Subscript[f, 1] == Subscript[f, 2] == \[Ellipsis] == Subscript[f, n + 1] == (2 c + I a (-2 k + m))/(2 c) && Subscript[g, 1] == Subscript[g, 2] == \[Ellipsis] == Subscript[g, n + 1] == (b (-2 k + u))/(2 c) && Subscript[h, 1] == Subscript[h, 2] == \[Ellipsis] == Subscript[h, n + 1] == (2 c + b (-2 k + u))/(2 c) && Subscript[i, 1] == Subscript[i, 2] == \[Ellipsis] == Subscript[i, n + 1] == ((-b) (-2 k + u))/(2 c) && Subscript[k, 1] == Subscript[k, 2] == \[Ellipsis] == Subscript[k, n + 1] == (2 c - b (-2 k + u))/(2 c) && Subscript[l, 1] == Subscript[l, 2] == \[Ellipsis] == Subscript[l, n + 1] == ((-I) a (-2 k + m) + b (-2 i + u))/ (2 c) && Subscript[m, 1] == Subscript[m, 2] == \[Ellipsis] == Subscript[m, n + 1] == (2 c - I a (-2 k + m) + b (-2 i + u))/(2 c) && Subscript[n, 1] == Subscript[n, 2] == \[Ellipsis] == Subscript[n, n + 1] == (I a (-2 k + m) + b (-2 i + u))/(2 c) && Subscript[o, 1] == Subscript[o, 2] == \[Ellipsis] == Subscript[o, n + 1] == (2 c + I a (-2 k + m) + b (-2 i + u))/(2 c) && Subscript[p, 1] == Subscript[p, 2] == \[Ellipsis] == Subscript[p, n + 1] == ((-I) a (-2 k + m) - b (-2 i + u))/(2 c) && Subscript[q, 1] == Subscript[q, 2] == \[Ellipsis] == Subscript[q, n + 1] == (2 c - I a (-2 k + m) - b (-2 i + u))/(2 c) && Subscript[r, 1] == Subscript[r, 2] == \[Ellipsis] == Subscript[r, n + 1] == (I a (-2 k + m) - b (-2 i + u))/(2 c) && Subscript[s, 1] == Subscript[s, 2] == \[Ellipsis] == Subscript[s, n + 1] == (2 c + I a (-2 k + m) - b (-2 i + u))/(2 c) && Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[u, Integers] && u > 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