Integrate[Sqrt[a + b Cosh[c z]]/(d + e Cosh[c z])^2, z] == (-((I (4 a d - b e) Sqrt[(a + b Cosh[c z])/(a + b)])/ (2 c (d - e) (d + e)^2 Sqrt[a + b Cosh[c z]]))) EllipticPi[(2 e)/(d + e), (I c z)/2, (2 b)/(a + b)] + ((1/(b c (d^2 - e^2))) (I Sqrt[-(1/(a + b))] (-a^2 + b^2) Sqrt[(b (1 + Cosh[c z]))/(-a + b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])) EllipticE[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - ((I Sqrt[(-b - b Cosh[c z])/(a - b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])/(Sqrt[-(1/(a + b))] c e (d + e))) EllipticF[I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - ((I b (2 d^2 + e^2) Sqrt[(b (1 + Cosh[c z]))/(-a + b)] Sqrt[(b - b Cosh[c z])/(a + b)] Csch[c z])/(2 Sqrt[-(1/(a + b))] c e (b d - a e) (-d^2 + e^2))) EllipticPi[((a + b) e)/((-b) d + a e), I ArcSinh[Sqrt[-(1/(a + b))] Sqrt[a + b Cosh[c z]]], (a + b)/(a - b)] - (e Sqrt[a + b Cosh[c z]] Sinh[c z])/(c (d - e) (d + e) (d + e Cosh[c z]))

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2002-12-18