I would not do this problem with complex analysis for two reasons:

1)The function is odd, usually we find the integral from to which is 1/2 of your desired integral if the function is even.

2)There are infinitely many poles for this function. And one of them is right on . Which means we have to avoid the origin somehow. Which is not looking good at all.

Maybe if the integral was we can do something nicer. Like for example take the contour with vertices at which has no poles and work from there.