Clearly, the case of this integral was introduced to the paper at the review stage, because it was of concern to the referee. Otherwise, it should have deserved a bit more of analysis from those mathematicians... The argument to abs is a sum of points on the unit circle, and its value can only depend on the difference
y-x. One approach is factoring out
exp(2*Pi*I*x), so that this integral remains:
The next observation is that this referee may have used Maple 17 or earlier. In Maple 18.01 this statement:
does not produce an answer within 1000s or so, while earlier versions return 0 in a few seconds. This huge time is spent mostly at computing one-sided limits at discontinuities of the primitive function returned by the method FTOC. And the previous behavior is recovered bypassing the discontinuity check as in:
Is this change of behavior a regression? Certainly, it would be of concern if it affects integrals that were correctly computed.
That paper raises though several points that would deserve further discussion here. For instance how far results from a CAS could be trusted, and strategies for checking results. Also, the clash between "open" science and closed source.