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http://functions.wolfram.com/01.07.21.1386.01
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Integrate[((a + b Cos[c z]^2)^\[Nu])^\[Beta] Cos[d z], z] ==
-((I 2^(\[Beta] \[Nu]) (a + ((1/4) b (1 + E^(2 I c z))^2)/E^(2 I c z))^
(\[Beta] \[Nu]) (E^(2 I d z) (d + 2 c \[Beta] \[Nu])
AppellF1[d/(2 c) - \[Beta] \[Nu], (-\[Beta]) \[Nu], (-\[Beta]) \[Nu],
1 + d/(2 c) - \[Beta] \[Nu], -((b E^(2 I c z))/
(2 a + b + 2 Sqrt[a (a + b)])),
-((b E^(2 I c z))/(2 a + b - 2 Sqrt[a (a + b)]))] -
(d - 2 c \[Beta] \[Nu]) AppellF1[-((d + 2 c \[Beta] \[Nu])/(2 c)),
(-\[Beta]) \[Nu], (-\[Beta]) \[Nu], 1 - d/(2 c) - \[Beta] \[Nu],
-((b E^(2 I c z))/(2 a + b + 2 Sqrt[a (a + b)])),
-((b E^(2 I c z))/(2 a + b - 2 Sqrt[a (a + b)]))])
((2 a + b + b Cos[2 c z])^\[Nu]/2^\[Nu])^\[Beta] Cos[d z])/
((1 + (b E^(2 I c z))/(2 a + b - 2 Sqrt[a (a + b)]))^(\[Beta] \[Nu])
(1 + (b E^(2 I c z))/(2 a + b + 2 Sqrt[a (a + b)]))^(\[Beta] \[Nu])
(2 a + b + b Cos[2 c z])^(\[Beta] \[Nu])))/
((1 + E^(2 I d z)) (d^2 - 4 c^2 \[Beta]^2 \[Nu]^2))
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<mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> β </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ </mo> <mi> β </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> β </mi> </mrow> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> c </mi> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <mi> β </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> b </mi> <mo> ⁢ </mo> 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Date Added to functions.wolfram.com (modification date)
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){kind=small}
