I'm having difficulty in using contour Integration to evaluate the integral:

where a>0.

Firstly I found that the function being integrated is an even function so this is equivalent to finding:

Also where Re dentoes the Real part of exp(ix) function. The integral in question is now:

so the complex function in question is:

which is holomorphic everywhere in the complex plane except for the points ai, and -ai. That is what I've found so far.

What contour do I have to use? Also how would you integrate it because I'm seeing there will be much difficulty integrating with the z being replaced by the suitable parameterization.

Thanks