David Bailey and Jonathan Borwein said in a talk yesterday that the most recent editions of both Maple and Mathematica give the nonsensical result $$\int_0^1\int_0^1|e^{2\pi ix}+e^{2\pi iy}|\,dx\,dy=0$$ I later verified this for Maple 17, entering int(int(abs(exp(2PiIx)+exp(2PiIy)),x=0..1),y=0..1).