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

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2002-12-18