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http://functions.wolfram.com/01.07.21.2419.01
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Integrate[Sin[d z] (a Sin[e z]^2 + b Sin[2 e z] + c 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 + (1 + E^(2 I e z))
(-2 I b (-1 + E^(2 I e z)) + c (1 + E^(2 I e z))))/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 + 2 I b - c) E^(2 I e z))/(a + c + 2 Sqrt[-b^2 + a c]),
-(((-a - 2 I b + c) E^(2 I e z))/(a + c - 2 Sqrt[-b^2 + a c]))] +
(d - 2 e \[Beta]) AppellF1[-((d + 2 e \[Beta])/(2 e)), -\[Beta],
-\[Beta], 1 - d/(2 e) - \[Beta], ((a + 2 I b - c) E^(2 I e z))/
(a + c + 2 Sqrt[-b^2 + a c]), -(((-a - 2 I b + c) E^(2 I e z))/
(a + c - 2 Sqrt[-b^2 + a c]))]))/
(E^(I d z) (1 + ((-a - 2 I b + c) E^(2 I e z))/
(a + c - 2 Sqrt[-b^2 + a c]))^\[Beta]
(1 + ((-a - 2 I b + c) E^(2 I e z))/(a + c + 2 Sqrt[-b^2 + a c]))^
\[Beta]))
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</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> c </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> <mo> + </mo> <mi> c </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> c </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> c </mi> 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Date Added to functions.wolfram.com (modification date)
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