Can integrals of the form $$ \int_{-\infty}^{\infty}{\exp\left(-\left[x - c\right]^{2}\right) \over 1 + x^{2}}\, {\rm d}x $$ be computed in closed form using contour integration (or any other technique)? If $c = 0$, the integral is $\pi{\rm e\ erfc}\left(1\right)$, but I'm interested in $c$ real and non-zero.

( In probability terms, the integrand is a product of normal and Cauchy densities. )