Integrate[Cos[b z^r + e]^m Cosh[c z^r + g], z] == -((1/r) ((-1)^m 2^(-1 - m) z Binomial[m, m/2] ((E^g Gamma[1/r, (-c) z^r])/((-c) z^r)^r^(-1) + Gamma[1/r, c z^r]/(E^g (c z^r)^r^(-1))) (1 - Mod[m, 2]))) - (1/r) (2^(-1 - m) z Sum[Binomial[m, k] ((E^(g - 2 I e k + I e m) Gamma[1/r, (-c + 2 I b k - I b m) z^r])/ ((-c + 2 I b k - I b m) z^r)^r^(-1) + (E^(-g - 2 I e k + I e m) Gamma[1/r, (c + 2 I b k - I b m) z^r])/ ((c + 2 I b k - I b m) z^r)^r^(-1) + (E^(g + 2 I e k - I e m) Gamma[1/r, (-c - 2 I b k + I b m) z^r])/ ((-c - 2 I b k + I b m) z^r)^r^(-1) + (E^(-g + 2 I e k - I e m) Gamma[1/r, (c - 2 I b k + I b m) z^r])/ ((c - 2 I b k + I b m) z^r)^r^(-1)), {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0

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