Calculate indefinite integral not defined in zero

Question

I would like to know any clue to calculate this integral: $\int_{R} \frac{1}{exp(x^{2})-a}$, where $a$ is a real parameter.

What methods could be used?

Answer 1

If $R$ does not contain the point where $e^{x^2}=a$, one may use the series expansion in terms of the error function: $$\int \frac{1}{e^{x^2}-a}dx = \int \frac{e^{-x^2}}{1-ae^{-x^2}}dx = \sum_{k\ge 0} a^k \int e^{-(1+k)x^2}dx = \frac12 \sum_{k\ge 0} a^k \sqrt{\frac{\pi}{1+k}} Erf(\sqrt{1+k}x). $$ Otherwise the integral is only defined as a principal value.