Evaluate $\frac{1}{2\sqrt{\pi b}}\int_{-\infty}^\infty\exp\left(-e^{x+a} - \frac{x^2}{4b}\right)dx$

Question

So, I'm trying to evaluate the integral $$ I(a,b) = \frac{1}{2\sqrt{\pi b}}\int_{-\infty}^\infty\exp\left(-e^{x+a} - \frac{x^2}{4b}\right)dx, $$ where $a$ and $b$ are real numbers with $b>0$. This seems like a simple enough looking integral that it would already exist in a table somewhere, but I can't seem to find it. The only thing I've figured out about it so far is that it satisfies $\partial_a^2 I = \partial_b I$ (there's essentially a heat kernel in the integrand). But the boundary conditions on that one are $I(a,0) = \exp(-e^a)$, $I(-\infty,b) = 1$, $I(\infty,b) = 0$, and this insight doesn't seem to lead anywhere. Anyone have any ideas?