# How to calculate this triple integral?

I was doing a physics problem and came across this integral which i need to solve

$$I(x,y,z)=\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi}\left(\frac{1}{8\pi^3}\right)\left(\frac{1-\cos(nx+my+pz))}{3-\cos(x)-\cos(y)-\cos(z)}\right)dx\,dy\,dz$$

What i want to know is that does this integral evaluate to any specific function $f(n,m,p)$, and if so then how to calculate it?

I tried Wolframalpha's triple integral calculator but that doesn't seem to work..

Are there any other online applications i can use to calculate such integrals?

• Are $n$ and $m$ and $p$ integers? – Greg.Paul May 21 '18 at 4:32
• Where did you get this integral? Where does it come from? The symmetry is nice and can maybe be exploited. – Greg.Paul May 21 '18 at 4:33
• @Greg.Paul Yes.They are actually coordinates of a node in a 3D grid and thus can have all possible integer values I saw this integral as a solution to a problem about calculating the equivalent resistance between nodes of an infinte N dimensional grid of unit resistors This is the case for N=3 or an infinite 3D grid of resistors Here is the link to the answer physics.stackexchange.com/questions/2072/… – Rutwik May 21 '18 at 4:34
• iopscience.iop.org/article/10.1088/0022-3719/8/11/002 – Count Iblis May 21 '18 at 4:36
• @CountIblis I cant open the PDF link.Requires some sort of a login – Rutwik May 21 '18 at 4:38