In my research, I ran into following types of improper integral

$\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$

with real parameters $a>0,b>0$.

Mathematica cannot evaluate them. It also seems that a definite integral of sort

$\int \cos (\sqrt{1+x^2})$, $\int \cosh (\sqrt{1+x^2})$ ,… etc

cannot be evaluated in mathematica, and i couldn't find them in the tables yet.

Could anyone help me how to solve these integrals?