We found some interesting bugs in Mathematica's integration software on this thread.
To wit, set
integral[m_,n_] = Integrate[Log[2+Cos[2Pi x]+Cos[2Pi y]] Cos[2Pi m x] Cos[2Pi n y],
{x, 0, 1}, {y, 0, 1}];
Then integral[1,1] should be $1/2-2/\pi$, but Mathematica 8.0.1 returns $1/2+2/\pi$. Values for other $m$ and $n$ are also wrong (see the question linked above), as can be quickly verified by replacing the "Integrate" command with "NIntegrate".
Curiously, if one changes the limits of integration to {x,-1/2,1/2} and {y,-1/2,1/2}, then the correct answers appear.
