I'm curious about the proper way to evaluate $$ \int_{-\infty}^{\infty} \frac{\cos \left(x-\frac{1}{x} \right)}{1+x^{2}} \, dx = \text{Re} \int_{-\infty}^{\infty} \frac{e^{i(x- \frac{1}{x})}}{1+x^{2}} \, dx$$ using contour integration.

If I let $f(z) = \frac{e^{i(z- \frac{1}{z})}}{1+z^{2}}$, there is an essential singularity at the origin.

So if I integrate around a closed semicircle in the upper half-plane, the contour goes right through the singularity.

Can you indent a contour around an essential singularity?