Integrate[Cos[z]^2/(Sqrt[a + b Sin[z]] (c + b Sin[z])^2), z] == (1/(4 b (a - c))) ((8 I a ((a - c) EllipticF[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sin[z]]], (a + b)/(a - b)] + c EllipticPi[(a + b)/(a - c), I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sin[z]]], (a + b)/(a - b)]) Sec[z] Sqrt[(b (1 + Sin[z]))/(-a + b)] Sqrt[(b - b Sin[z])/(a + b)])/ (b^2 Sqrt[-(1/(a + b))] (a - c)) - (I (Cos[z] + Cos[3 z]) (2 (a - b) (a - c) EllipticE[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sin[z]]], (a + b)/(a - b)] + 2 (a - c) (b + c) EllipticF[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sin[z]]], (a + b)/(a - b)] - (b^2 - 2 c^2) EllipticPi[(a + b)/(a - c), I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Sin[z]]], (a + b)/(a - b)]) Sec[z]^2 Sec[2 z] Sqrt[(b (1 + Sin[z]))/(-a + b)] Sqrt[(b - b Sin[z])/(a + b)])/(b^2 Sqrt[-(1/(a + b))] (a - c)) + (6 b EllipticPi[(2 b)/(b + c), (1/4) (Pi - 2 z), (2 b)/(a + b)] Sqrt[(a + b Sin[z])/(a + b)])/((b + c) Sqrt[a + b Sin[z]]) - (4 Cos[z] Sqrt[a + b Sin[z]])/(c + b Sin[z]))

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<mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; 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2002-12-18