How to evaluate $\int_{-\infty}^\infty \frac{\operatorname{sech}x}{1+x^2}dx$?

Question

I am trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\operatorname{sech}x}{1+x^2}dx\,.$$ This integral shows up in my physics research, and I see that a contour can't be so easily used since $\operatorname{sech}x$ has poles all along the imaginary axis. Any hints?