Integrate[Sqrt[a + b Sinh[c z]] Coth[c z]^2, z] == (1/(4 c)) ((1/(a Sqrt[1/(-a + I b)] b)) (6 (I a (a + I b) EllipticE[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] + b (a EllipticF[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] + I b EllipticPi[1 - (I b)/a, I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)])) Sech[c z] Sqrt[(b - I b Sinh[c z])/(I a + b)] Sqrt[(b + I b Sinh[c z])/((-I) a + b)]) - 4 Coth[c z] Sqrt[a + b Sinh[c z]] + (8 I a EllipticF[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)])/Sqrt[a + b Sinh[c z]] + (1/Sqrt[a + b Sinh[c z]]) (6 ((I a + b) EllipticE[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] - I a EllipticF[(1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))]) Sqrt[(a + b Sinh[c z])/(a - I b)]) + (2 b EllipticPi[2, (1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sqrt[(a + b Sinh[c z])/(a - I b)])/Sqrt[a + b Sinh[c z]])

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2002-12-18