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; }

 Sech

 http://functions.wolfram.com/01.24.21.0270.01

 Input Form

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