Triple Integration

Calculus Level 5

\[\large \int_{0}^{2\pi} \int_{0}^{2\pi} \int_{0}^{2\pi} \dfrac{ dx \; dy \; dz}{ 1- \frac 13 ( \cos x + \cos y + \cos z) }\]

If the closed form of the integral above can be expressed as:

\[\dfrac{\sqrt{A}}{4} \Gamma \left( \dfrac{B}{24}\right)\Gamma\left(\dfrac{C}{24}\right) \Gamma \left(\dfrac{D}{24}\right) \Gamma \left(\dfrac{E}{24}\right)\]

Find \(A+B+C+D+E\).

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