Integrate[1/(Sqrt[(a + b Sinh[e z]^2) (c + d Sinh[e z]^2)] (f + g Sinh[e z]^2)), z] == (Sqrt[a + b Sinh[e z]^2] Sqrt[c + d Sinh[e z]^2] (-((I a^2 c g Cosh[e z] Sqrt[(2 a - b + b Cosh[2 e z])/a] Sqrt[(a (2 c - d + d Cosh[2 e z]))/(c (2 a - b + b Cosh[2 e z]))] EllipticPi[(b f - a g)/((-a) f + b f), I ArcSinh[(Sqrt[2 - (2 b)/a] Sinh[e z])/ Sqrt[(2 a - b + b Cosh[2 e z])/a]], (b c - a d)/((-a) c + b c)])/ Sqrt[(a Cosh[e z]^2)/(2 a - b + b Cosh[2 e z])]) + b Sqrt[1 - b/a] ((-b) c + a d) f Sqrt[(c Coth[e z]^2)/(c - d)] Sqrt[(c (2 a - b + b Cosh[2 e z]) Csch[e z]^2)/(b c - a d)] Sqrt[(a (2 c - d + d Cosh[2 e z]) Csch[e z]^2)/((-b) c + a d)] EllipticF[ArcSin[Sqrt[(a (2 c - d + d Cosh[2 e z]) Csch[e z]^2)/ (-2 b c + 2 a d)]], (b c - a d)/(a c - a d)] Sinh[e z]^2 Tanh[e z]))/ (a Sqrt[1 - b/a] c e f ((-b) f + a g) Sqrt[2 a - b + b Cosh[2 e z]] Sqrt[2 c - d + d Cosh[2 e z]] Sqrt[(a + b Sinh[e z]^2) (c + d Sinh[e z]^2)])

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2002-12-18