Integrate[Sqrt[a + b Sinh[c z]] Csch[c z]^2, z] == -((Cosh[c z] + Cosh[3 c z]) Csch[c z] Sech[c z]^2 Sech[2 c z] (-2 I b^2 EllipticPi[1 - (I b)/a, I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] Sinh[c z] Sqrt[-((b (-I + Sinh[c z]))/(a + I b))] Sqrt[-((b (I + Sinh[c z]))/(a - I b))] Sqrt[a + b Sinh[c z]] + a (4 ((-I) a + b) EllipticE[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] Sinh[c z] Sqrt[-((b (-I + Sinh[c z]))/(a + I b))] Sqrt[-((b (I + Sinh[c z]))/(a - I b))] Sqrt[a + b Sinh[c z]] + b (-4 EllipticF[I ArcSinh[Sqrt[1/(-a + I b)] Sqrt[a + b Sinh[c z]]], (a - I b)/(a + I b)] Sinh[c z] Sqrt[-((b (-I + Sinh[c z]))/ (a + I b))] Sqrt[-((b (I + Sinh[c z]))/(a - I b))] Sqrt[a + b Sinh[c z]] + 2 Sqrt[1/(-a + I b)] Cosh[c z] (2 Cosh[c z] (a + b Sinh[c z]) + b EllipticPi[2, (1/4) (Pi - 2 I c z), -((2 I b)/(a - I b))] Sinh[c z] Sqrt[(a + b Sinh[c z])/(a - I b)])))))/ (8 a Sqrt[1/(-a + I b)] b c Sqrt[a + b Sinh[c z]])

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2002-12-18