Integrate[Sin[b Sqrt[z]] Cosh[c z]^v, z] == (1/b^2) 2^(1 - v) Binomial[v, v/2] (1 - Mod[v, 2]) ((-b) Sqrt[z] Cos[b Sqrt[z]] + Sin[b Sqrt[z]]) + 2^(-1 - v) Sum[Binomial[v, k] ((1/(2 I c k - I c v)^(3/2)) (b Sqrt[2 Pi] ((-I) Sinh[b^2/(4 (2 c k - c v))] FresnelC[(-b + 2 I c (2 k - v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[ I c (2 k - v)])] - FresnelS[(-b + 2 I c (2 k - v) Sqrt[z])/ (Sqrt[2 Pi] Sqrt[I c (2 k - v)])] Cosh[b^2/(4 (2 c k - c v))]) + 2 Sqrt[2 I c k - I c v] Cos[b Sqrt[z] - I c (2 k - v) z]) + (1/(2 I c k - I c v)^(3/2)) ((-b) Sqrt[2 Pi] (I Sinh[b^2/(4 (2 c k - c v))] FresnelC[(b + 2 I c (2 k - v) Sqrt[z])/(Sqrt[2 Pi] Sqrt[ I c (2 k - v)])] + FresnelS[(b + 2 I c (2 k - v) Sqrt[z])/ (Sqrt[2 Pi] Sqrt[I c (2 k - v)])] Cosh[b^2/(4 (2 c k - c v))]) - 2 Sqrt[2 I c k - I c v] Cos[b Sqrt[z] + I c (2 k - v) z])), {k, 0, Floor[(1/2) (-1 + v)]}] /; Element[v, Integers] && v > 0

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2002-12-18