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No, there is no closed form. However, $\displaystyle\int^\infty_0 e^{-a x^2} \cosh (bx)dx=\frac12\exp\bigg(\frac{b^2}{4a}\bigg)\sqrt\frac\pi a~$ should make for a decent lower limit. Though, depending on the ratio between a to b, its value can be up to several $/$ many times smaller than that of the original. In any case, asymptotic approximation seems to be the way to go, perhaps by adding a second approximating function for the integrand, for small values of x, since the one mentioned above works better for larger values of the variable, as $x\to\infty$. Hope this helps.