Consider $$ I = \int_{-L}^L f(x)dx, $$ where $f(x)$ is real-valued and analytic on $[-L,L]$, but it has a pole in the complex plane whose real part lies in $[-L,L]$. Call it $z_0$, and assume it is a simple pole. If the imaginary part of $z_0$ is small compared to the real part, then $f(x)$ is sharply peaked in the vicinity of $x_0=\mathrm{Re}\{z_0\}$. If this peak is sufficiently sharp and $f(x)$ is slowly varying elsewhere, then basic adaptive quadrature methods (e.g., recursive Simpson or Gauss-Legendre) will spend considerable effort attempting to resolve the peak.
An obvious thing to try is to approximate the value of $x_0$ and break the integral up $$ I = I_-+I_+ = \int_{-L}^{x_0}f(x)dx+\int_{x_0}^Lf(x)dx $$ But doing so only pushes the problem to the endpoints of the integration domain instead of the interior. I wonder if there is a standard change of variables to alleviate this behavior, though?
Another option could be to try to evaluate $I$ by means of contour integration in the complex plane and use of the residue theorem. However, this will not fly if the software implementation of $f$ is only defined for real arguments.
Are there other specialized quadrature methods for these kinds of "peaky" integrands? Ideally I'd like to make no further assumptions on $f$ execpt that it's real, analytic, and slowly varying away from $x_0$. But I am also open to answers that require more assumptions on $f$ if there is no general answer.
