Convergence of $\int_{-\infty}^{\infty} \frac{e^{-\frac 12 (x^2 - (1-a)^2)}}{\sqrt{a^2 + x^2}\sqrt{(1-a)^2 + x^2}}dx, \ a<0$

Question

I came across the following integral $$\int_{-\infty}^{\infty} \frac{e^{-\frac 12 (x^2 - (1-a)^2)}}{\sqrt{a^2 + x^2}\sqrt{(1-a)^2 + x^2}}dx, \quad a<0.$$

Can it be proven that the integral converge for every $a<0$?

EDIT

Just a thought, kinda looks like a chi distribution, maybe.