Can anyone help me to solve the following problem? I want to compute the integral
\begin{equation}\label{eq:q0} \lim_{\varepsilon\rightarrow 0}\,\int_{-\infty}^{+\infty} dx \, \frac{1}{x^2-a^2+i\varepsilon} \, \frac{1}{1+\exp[2(x^2-b^2)]}\,, \end{equation}
(edited in response to the comment)
which has poles at $x= \pm a\mp i\varepsilon$ and $x= \pm\sqrt{(2 b^2 + i π + 2 i n π)/2}$, $n\in\mathbb{Z}$.
Computing the residues for $x= \pm a\mp i\varepsilon$ is easy, but how can I compute the other residues? I considered making the substitution $z=\exp[2(x^2-b^2)]$, $dz=\exp[2(x^2-b^2)]4xdx$, but the resulting integral still seems to be too complicated to be solvable.