Pensieve header: a Fubini counterexample from Kuperberg’s FaceBook.

f=((x-y)/(((1+(((x-y))^2)))^2))

Plot3D[f, {x,0,3},{y,0,3}, PlotRange -> All, PlotPoints->100]

Assuming[y>=0,Integrate[f,{x,0,∞}]]

Integrate[(1/(2+2 (y^2))),{y,0,∞}]

Assuming[x>=0,Integrate[f,{y,0,∞}]]

Integrate[-(1/(2+2 (x^2))),{x,0,∞}]