@vv If it helps you can transform the integral into a sum of the form
where Im(c)>0 if Im(b)<0 using the residue theorem. (The arc in the upper plane should vanish when epsilon>0 and t>0)
Do you post the solution here if you uploaded it to the application center (where do I find it?)
I'm still suprised about the discontinuities at t=0 and t=2 for example. If Im(b)<0 there is no singularity on the axis, so why do you need to calculate the integral by CPV anyhow?
Apparently when t=2 in this sum you get 1^k=1 and a badly blowing up series if epsilon <= 1. So in that sense I'm surprised why it is at all finite (as you say as a CPV at t=2\IZ) just with discontinuities.