I'm working on a physics problem which needs the analytic formula for the following integral:
$$\int_{0}^{V_1}\int_{0}^{V_2} \frac{4 p^4}{\cos(2ap/\hbar)\left(2p^2z_1z_2-\frac{z_1^2z_2^2}{2}\right)+p^2(z_1^2+z_2^2)+\frac{1}{2}z_1^2z_2^2-\sin(2ap/\hbar)pz_1z_2(z_1+z_2)+4p^4}dz_1dz_2 $$
the denominator is symmetric in $z_1$ and $z_2$ so I think there may exist an analytic formula for it. Mathematica is able to work out one integral and gives me something in terms of $\arctan$ but it fails to do the second. The first indefinite integral (dropping the $\hbar$) evaluates to this:
$$-\frac{4 p^3 \tan ^{-1}\left(\frac{p \left(-z_1 \sin (a p)-z_2 \sin (a p)+2 p \cos (a p)\right)}{p \left(z_2 \cos (a p)+2 p \sin (a p)\right)+z_1 \left(p \cos (a p)-z_2 \sin (a p)\right)}\right)}{z_2^2 \sin ^2(a p)-p z_2 \sin (2 a p)+2 p^2}$$
Is there any possible substitution or coordinate transformation to work this out?