Is there a known form for indefinite Gaussian integrals of the form
$$\int dx\, x^{-n}\exp(-(x-b)^2) $$
where $n$ is a positive integer and $b$ is some constant? Mathematica cannot solve integrals of this type, and for some reason I cannot find any terms like this in any integral tables.
Functions like these tend not to have antiderivatives expressible in closed form in terms of elementary functions.
In the special case $b = 0$, we can write antiderivatives of these functions in terms of the exponential integral $$\operatorname{Ei}_1(x) := \int_1^\infty t^{-1} \exp(-t x) \,dt$$ (for odd $n$) and the error function, $$\operatorname{erf}(x) := \frac{2}{\sqrt\pi} \int_0^x \exp(-t^2) \,dt$$ (for even $n$), but that doesn't provide much insight in the sense that those functions are defined as antiderivatives of functions similar to those in the question.
The first few values are: $$\begin{array}{cc} n & \displaystyle{\int x^{-n} \exp(-x^2) \,dx} \\ \hline 1 & -\frac{1}{2}\operatorname{Ei}_1(x^2) + C\\ 2 & x^{-1} \exp(-x^2) - \sqrt{\pi}\operatorname{erf}(x) + C \\ 3 & \frac{1}{2} \left[-x^{-3} \exp(-x^2) + \operatorname{Ei}_1(x^2)\right] + C \end{array} .$$