Gaussian type integral with inverse square root

Question

Hi,

I have encountered an integral of the following type in an engineering application:

$\int_{-\infty}^\infty dx \frac{1}{\sqrt{x^2+a^2}}\exp(-x^2/2+i x b)$,

where $a$ and $b$ are real ($a$ could be zero). Is it possible to solve this integral analytically?

Thanks for the help and comments.