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http://functions.wolfram.com/03.04.21.0040.01
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Integrate[z^(\[Alpha] - 1) Sinh[b + a z^r] BesselK[\[Nu], a z^r], z] ==
(2^(-1 - \[Nu]) Pi z^\[Alpha] Csc[Pi \[Nu]]
((4^\[Nu] a z^r Cosh[b] HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2,
1/2 + \[Alpha]/(2 r) - \[Nu]/2}, {3/2, 1 - \[Nu], 3/2 - \[Nu],
3/2 + \[Alpha]/(2 r) - \[Nu]/2}, a^2 z^(2 r)])/
((r + \[Alpha] - r \[Nu]) Gamma[1 - \[Nu]]) +
(1/((\[Alpha] - r \[Nu]) Gamma[1 - \[Nu]]))
(4^\[Nu] HypergeometricPFQ[{1/4 - \[Nu]/2, 3/4 - \[Nu]/2,
\[Alpha]/(2 r) - \[Nu]/2}, {1/2, 1/2 - \[Nu], 1 - \[Nu],
1 + \[Alpha]/(2 r) - \[Nu]/2}, a^2 z^(2 r)] Sinh[b]) +
(1/Gamma[1 + \[Nu]]) ((a z^r)^(2 \[Nu])
(-((1/(r + \[Alpha] + r \[Nu])) (a z^r Cosh[b] HypergeometricPFQ[
{3/4 + \[Nu]/2, 5/4 + \[Nu]/2, 1/2 + \[Alpha]/(2 r) + \[Nu]/2},
{3/2, 3/2 + \[Alpha]/(2 r) + \[Nu]/2, 1 + \[Nu], 3/2 + \[Nu]},
a^2 z^(2 r)])) - (1/(\[Alpha] + r \[Nu]))
(HypergeometricPFQ[{1/4 + \[Nu]/2, 3/4 + \[Nu]/2, \[Alpha]/(2 r) +
\[Nu]/2}, {1/2, 1 + \[Alpha]/(2 r) + \[Nu]/2, 1/2 + \[Nu],
1 + \[Nu]}, a^2 z^(2 r)] Sinh[b])))))/(a z^r)^\[Nu]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mi> α </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> α </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> 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Date Added to functions.wolfram.com (modification date)
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