Integrate[1/(Sqrt[(a + b Cosh[e z]^2) (c + d Cosh[e z]^2)] (f + g Cosh[e z]^2)), z] == -(I Sqrt[a + b Cosh[e z]^2] Sqrt[c + d Cosh[e z]^2] ((a g Cosh[e z] Sqrt[(2 a + b + b Cosh[2 e z])/(a + b)] Sqrt[((a + b) (2 c + d + d Cosh[2 e z]))/ ((c + d) (2 a + b + b Cosh[2 e z]))] EllipticPi[ ((-b) f + a g)/(a (f + g)), I ArcSinh[(Sqrt[2] Sqrt[a/(a + b)] Sinh[e z])/ Sqrt[(2 a + b + b Cosh[2 e z])/(a + b)]], ((-b) c + a d)/ (a (c + d))])/((a/(a + b))^(3/2) (f + g) Sqrt[((a + b) Cosh[e z]^2)/(2 a + b + b Cosh[2 e z])]) - (I b (b c - a d) Sqrt[((c + d) (2 a + b + b Cosh[2 e z]) Csch[e z]^2)/ (b c - a d)] Sqrt[((a + b) (2 c + d + d Cosh[2 e z]) Csch[e z]^2)/ ((-b) c + a d)] EllipticF[ArcSin[ Sqrt[((a + b) (2 c + d + d Cosh[2 e z]) Csch[e z]^2)/ (-2 b c + 2 a d)]], (b c - a d)/(a c + b c)] Sinh[2 e z])/ (2 (a + b) c Sqrt[((c + d) Coth[e z]^2)/c])))/ (e ((-b) f + a g) Sqrt[(a + b Cosh[e z]^2) (c + d Cosh[e z]^2)] Sqrt[2 a + b + b Cosh[2 e z]] Sqrt[2 c + d + d Cosh[2 e z]])

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2002-12-18