Notice that your integral is the real part of , so it suffices to compute this one.
Now, let .
The integral you want is obtained by integrating along the real axis: .
The integral you know arises when , which means (for instance) (I took a square root of ).
So you should prove that integrating along gives the same result as integrating along (this is the bisector of the north-east quarter plane).
Given , you could consider the contour that goes from 0 to on , then follows a piece of circle into , and goes back to 0 along the radius " ". You know what the integral along this contour is equal to (as you said, there is no singularity). Then the idea is to make and prove that the integration on the piece of circle tends to 0. This would give you the equality you need.
Write down what is the integral of along the three different parts of the contour (the two radii and the piece of arc) to try to complete the proof. If you're still stuck, tell us what you found.