Can anyone see a solution for varitations of
$$\int \frac{1}{x+b}\exp(-(x\pm b)^2)dx$$
I recognize that the term $1/(x+b)$ could be rewritten for the region $|x|<b$
$$\int \sum_{i=1}^\infty \frac{(-x)^i}{b^i} \exp(-(x \pm b)^2) \textrm{d}x $$
The series can be modified to an arbitrarily large convergence domain by changing the center of the expansion
$$\int \sum_{i=1}^\infty \frac{(x+L)^i}{(L -b)^i} \exp(-(x \pm b)^2) \textrm{d}x $$
But I can't go much further with this approach.
I expect (not required) to have some solution of the form $Q_1(b,x) \textrm{erfc}(x \pm b) + Q_2(b,x) e^{-(x\pm b)^2}$ where $Q_1(x),Q_2(x)$, or a series in terms of $Q_k(x) \gamma_k(x)$ (incomplete gamma function), and the $Q$ functions are rational. Valid for specific intervals that will cover the positive real domain. Or simply an exact series that converges quikly for large $b$.
Thank you
