Although there is no closed form in terms of elementary functions, $$\int^\infty_0 e^{-a x^2} \cosh (bx)dx=\frac12\exp\bigg(\frac{b^2}{4a}\bigg)\sqrt\frac\pi a$$ should make for a decent lower limit. Though, depending on the ratio between a and b, its value
can be up to several $/$ many times smaller than that of the original. In any case, the asymptotic
approximation can be significantly improved, 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.