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http://functions.wolfram.com/01.07.21.1187.01
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Integrate[Cos[c z]^2 Sqrt[Cos[a z]], z] ==
(4 (a^2 - 16 c^2) Sqrt[1 + E^(2 I a z)] EllipticE[(a z)/2, 2] +
(I Sqrt[2] a Sqrt[E^((-I) a z) + E^(I a z)]
((a - 4 c) Hypergeometric2F1[-(1/4) - c/a, -(1/2), 3/4 - c/a,
-E^(2 I a z)] + (a + 4 c) E^(4 I c z) Hypergeometric2F1[-(1/4) + c/a,
-(1/2), 3/4 + c/a, -E^(2 I a z)]))/E^(2 I c z))/
(4 (a^3 - 16 a c^2) Sqrt[1 + E^(2 I a z)])
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⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mi> c </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mi> E </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ❘ </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> c </mi> <mo> ⁢ 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<mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mi> c </mi> <mi> a </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[FractionBox["c", "a"], "-", FractionBox["1", "4"]]], Hypergeometric2F1], ",", TagBox[RowBox[List["-", FractionBox["1", "2"]]], Hypergeometric2F1]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1], ";", TagBox[TagBox[TagBox[RowBox[List[FractionBox["c", 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type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 4 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <ci> c </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 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Date Added to functions.wolfram.com (modification date)
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){kind=small}
