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http://functions.wolfram.com/01.24.21.0059.01
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Integrate[z^n Cos[b z]^m Sech[c z], z] ==
2^(1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) n! E^(c z)
Sum[(((-1)^j z^(-j + n))/((n - j)! c^(j + 1))) 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}] + 2^(1 - m) n!
Sum[Binomial[m, k] (E^((c - I b (-2 k + m)) z)
Sum[(((-1)^j z^(-j + n))/((n - j)! (c - I b (-2 k + m))^(j + 1)))
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^((c + I b (-2 k + m)) z) Sum[(((-1)^j z^(-j + n))/
((n - j)! (c + I b (-2 k + m))^(j + 1))) 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}]), {k, 0, Floor[(1/2) (-1 + m)]}] /;
Subscript[b, 1] == Subscript[b, 2] == \[Ellipsis] == Subscript[b, n + 1] ==
1/2 && Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] ==
Subscript[c, n + 1] == (c - I b (-2 k + m))/(2 c) &&
Subscript[d, 1] == Subscript[d, 2] == \[Ellipsis] == Subscript[d, n + 1] ==
(c + I b (-2 k + m))/(2 c) && Element[n, Integers] && n >= 0 &&
Element[m, Integers] && m >= 0
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<apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <plus 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<ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> <apply> <times /> <apply> <plus 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<ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> m </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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