I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when $x=\frac{1}{2}(n+\frac{1}{2})\pi$ but the $\cosh(x-a)$ term relaxes these singularities exponentially as $x$ goes to infinity.
If it does exist, then I'm left with the problem of computing it. This is for a complex variables class, so I was using residue methods. However, there are countably many simple poles along the real axis as well as for each $z=i(n+\frac{1}{2})\pi +a$ in the complex plane, so I don't even know what contour to use. It looks like rectangles and semicircles are out. Any suggestions?