If $b=0$, then it reduces to $\int_{0}^{\infty} \frac{e^{-(x+a/x)}}{x} dx$, which is $2K_0(2\sqrt{a})$, where $K_0(x)$ is the zero-order modified Bessel function of the second kind.
However, if $b\neq0$, I do not know how to evaluate the integral or relate it to $K_n(x)$ or some other known function. Any help is much appreciated!