Integrate[1/((d + e Sin[c z])^3 Sqrt[a + b Sin[c z]^2]), z] == -(((a^2 e^4 (2 d^2 + e^2) + a b e^2 (5 d^4 + 2 d^2 e^2 - e^4) + b^2 (6 d^6 - 5 d^4 e^2 + 2 d^2 e^4)) ArcTanh[(Sqrt[2 + (2 b d^2)/(a e^2)] Cos[c z])/(Sqrt[1 - d^2/e^2] Sqrt[(2 a + b - b Cos[2 c z])/a])] Sqrt[(2 a + b - b Cos[2 c z])/a])/ (2 c Sqrt[1 - d^2/e^2] Sqrt[1 + (b d^2)/(a e^2)] e (d^2 - e^2)^2 (b d^2 + a e^2)^2 Sqrt[2 a + b - b Cos[2 c z]])) + (3 d e^2 (2 b d^2 + a e^2 - b e^2) Sqrt[2 a + b - b Cos[2 c z]] EllipticE[c z, -(b/a)])/(2 c (d^2 - e^2)^2 (b d^2 + a e^2)^2 Sqrt[(2 a + b - b Cos[2 c z])/a]) - (d (4 b d^2 + 3 a e^2 - b e^2) Sqrt[(2 a + b - b Cos[2 c z])/a] EllipticF[c z, -(b/a)])/(2 c (d^2 - e^2)^2 (b d^2 + a e^2) Sqrt[2 a + b - b Cos[2 c z]]) + ((a^2 e^4 (2 d^2 + e^2) + a b e^2 (5 d^4 + 2 d^2 e^2 - e^4) + b^2 (6 d^6 - 5 d^4 e^2 + 2 d^2 e^4)) Sqrt[(2 a + b - b Cos[2 c z])/a] EllipticPi[e^2/d^2, c z, -(b/a)])/(2 c d (d^2 - e^2)^2 (b d^2 + a e^2)^2 Sqrt[2 a + b - b Cos[2 c z]]) + (4 Sqrt[2] e^3 Cos[c z] (-d + e Sin[c z])^5 (d + e Sin[c z])^3 (-28 a b d^4 - 14 b^2 d^4 - 16 a^2 d^2 e^2 + 8 a b d^2 e^2 + 8 b^2 d^2 e^2 + 4 a^2 e^4 + 2 a b e^4 + 2 b (7 b d^4 + 4 a d^2 e^2 - 4 b d^2 e^2 - a e^4) Cos[2 c z] - 3 (4 a + 3 b) d e (2 b d^2 + a e^2 - b e^2) Sin[c z] + 6 b^2 d^3 e Sin[3 c z] + 3 a b d e^3 Sin[3 c z] - 3 b^2 d e^3 Sin[3 c z]))/(c (d^2 - e^2)^2 (b d^2 + a e^2)^2 Sqrt[2 a + b - b Cos[2 c z]] (2 d^2 - e^2 + e^2 Cos[2 c z])^5)

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</mfrac> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ; </mo> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> d </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msqrt> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> e </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 14 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 28 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 3 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> d </mi> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msqrt> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> &#10072; </mo> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mi> a </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> </mfrac> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msqrt> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 2 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mi> a </mi> </mfrac> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> e </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> d </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> e </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msqrt> <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> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> e </ci> <apply> <sin /> 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<cn type='integer'> 6 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> e </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticPi </ci> <apply> <times /> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> d </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> d </ci> <cn 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<apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> d </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> <ci> b </ci> <apply> <power /> <ci> e </ci> <cn type='integer'> 3 </cn> </apply> <apply> <sin /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> e </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> b </ci> <apply> <power /> <ci> e </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 7 </cn> <ci> b </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> 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type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> d </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> e </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> 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2002-12-18