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http://functions.wolfram.com/01.07.21.2405.01
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Integrate[Sin[d z] (a Sin[e z]^2 + b Cos[e z]^2)^\[Beta], z] ==
(-(1/(d^2 - 4 e^2 \[Beta]^2)))
((2^(-1 - 2 \[Beta]) (((-a) (-1 + E^(2 I e z))^2 + b (1 + E^(2 I e z))^2)/
E^(2 I e z))^\[Beta] (E^(2 I d z) (d + 2 e \[Beta])
AppellF1[d/(2 e) - \[Beta], -\[Beta], -\[Beta], 1 + d/(2 e) - \[Beta],
((a - b) E^(2 I e z))/(a + b + 2 Sqrt[a b]), ((a - b) E^(2 I e z))/
(a + b - 2 Sqrt[a b])] + (d - 2 e \[Beta])
AppellF1[-((d + 2 e \[Beta])/(2 e)), -\[Beta], -\[Beta],
1 - d/(2 e) - \[Beta], ((a - b) E^(2 I e z))/(a + b + 2 Sqrt[a b]),
((a - b) E^(2 I e z))/(a + b - 2 Sqrt[a b])]))/
(E^(I d z) (1 + ((-a + b) E^(2 I e z))/(a + b - 2 Sqrt[a b]))^\[Beta]
(1 + ((-a + b) E^(2 I e z))/(a + b + 2 Sqrt[a b]))^\[Beta]))
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AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mfrac> <mo> - </mo> <mi> β </mi> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mi> β </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> ; </mo> <mrow> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> e </mi> </mrow> </mfrac> <mo> - </mo> <mi> β </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> ( 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){kind=small}
