Integrate[1/(Sqrt[(a + b Sin[e z]^2) (c + d Sin[e z]^2)] (f + g Sin[e z]^2)), z] == (Sqrt[a + b Sin[e z]^2] Sqrt[c + d Sin[e z]^2] ((a^2 c g Cos[e z] Sqrt[(2 a + b - b Cos[2 e z])/a] Sqrt[(a (2 c + d - d Cos[2 e z]))/(c (2 a + b - b Cos[2 e z]))] EllipticPi[(b f - a g)/(a f + b f), ArcSin[(Sqrt[(2 a + 2 b)/a] Sin[e z])/Sqrt[(2 a + b - b Cos[2 e z])/ a]], (b c - a d)/(a c + b c)])/ Sqrt[(a Cos[e z]^2)/(2 a + b - b Cos[2 e z])] + b Sqrt[(a + b)/a] ((-b) c + a d) f Sqrt[-((c Cot[e z]^2)/(c + d))] Sqrt[(c (2 a + b - b Cos[2 e z]) Csc[e z]^2)/(b c - a d)] Sqrt[(a (2 c + d - d Cos[2 e z]) Csc[e z]^2)/((-b) c + a d)] EllipticF[ArcSin[Sqrt[(a (2 c + d - d Cos[2 e z]) Csc[e z]^2)/ (-2 b c + 2 a d)]], ((-b) c + a d)/(a (c + d))] Sin[e z]^2 Tan[e z]))/(a Sqrt[(a + b)/a] c e f ((-b) f + a g) Sqrt[2 a + b - b Cos[2 e z]] Sqrt[2 c + d - d Cos[2 e z]] Sqrt[(a + b Sin[e z]^2) (c + d Sin[e z]^2)])

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</mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> d </mi> <mo> - </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </msqrt> 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<mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> </mrow> <mo> + </mo> <mi> d </mi> <mo> - </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> c </mi> </mrow> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> 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2002-12-18