Triple Integration

This is basically one of the three famous Watson triple integrals $\int_0^{2\pi}\int_0^{2\pi}\int_0^{2\pi} \frac{1}{1 - \frac13(\cos x + \cos y + \cos z)}\,dx\,dy\,dz \; = \; 24\pi^3I_3$ where $I_3 \; = \; \frac{1}{\pi^3}\int_0^\pi \int_0^\pi \int_0^\pi \frac{1}{3 - \cos x - \cos y - \cos z}\,dx\,dy\,dz \; =\; \frac{1}{16\sqrt{6}\pi^3}\Gamma\big(\tfrac{1}{24}\big)\Gamma\big(\tfrac{5}{24}\big)\Gamma\big(\tfrac{7}{24}\big)\Gamma\big(\tfrac{11}{24}\big)$ The proof is in the original 1939 paper, but some details can be found here .

This makes this integral equal to $\frac{\sqrt{6}}{4}\Gamma\big(\tfrac{1}{24}\big)\Gamma\big(\tfrac{5}{24}\big)\Gamma\big(\tfrac{7}{24}\big)\Gamma\big(\tfrac{11}{24}\big)$ and so the desired answer is $6+1+5+7+11 = \boxed{30}$ .