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http://functions.wolfram.com/01.10.21.0056.01
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Integrate[z^n Cos[b z]^m Csc[c z], z] ==
I 2^(1 - m) E^(I c z) Binomial[m, m/2] n! (-1 + Mod[m, 2])
Sum[(1/(-j + n)!) (-1)^j z^(-j + n) (I c)^(-1 - j)
HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1],
1}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]},
E^(2 I c z)], {j, 0, n}] - I 2^(1 - m) E^(I c z) n!
Sum[Binomial[m, k] (E^(I b (-2 k + m) z) Sum[(1/(-j + n)!) (-1)^j
z^(-j + n) (I b (-2 k + m) + I c)^(-1 - j) HypergeometricPFQ[
{Subscript[b, 1], \[Ellipsis], Subscript[b, j + 1], 1},
{1 + Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, j + 1]},
E^(2 I c z)], {j, 0, n}] + Sum[(1/(-j + n)!) (-1)^j z^(-j + n)
((-I) b (-2 k + m) + I c)^(-1 - j) HypergeometricPFQ[
{Subscript[c, 1], \[Ellipsis], Subscript[c, j + 1], 1},
{1 + Subscript[c, 1], \[Ellipsis], 1 + Subscript[c, j + 1]},
E^(2 I c z)], {j, 0, n}]/E^(I b (-2 k + m) z)),
{k, 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 + b (-2 k + m))/(2 c) &&
Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, n + 1] ==
(c - b (-2 k + m))/(2 c) && Element[n, Integers] && n >= 0 &&
Element[m, Integers] && m > 0
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<apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <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 /> <ci> c </ci> <apply> <times /> <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> <cn type='integer'> 1 </cn> </list> <list> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> b </ci> <apply> 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<apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> c </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </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 /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </list> <list> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <ci> … </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </list> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <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> <ci> ℕ </ci> </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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