Solution for $\int_0^\infty e^{-(ct)^\alpha} \cos(x t) dt$

Question

I'm trying to evaluate the integral: $\int_0^\infty e^{-(ct)^\alpha} \cos(x t) dt$, where $\alpha$ and $c$ are parameters.

This integral arises from trying to solve for the probability density for a symmetric $\alpha$-stable probability distribution.

Can this integral be expressed in terms of some special functions, or at least does it have a ready-made numerical solution method?