Hot Integral - 25

Calculus Level 5

00001wxyz(w+x+y+z+1w+1x+1y+1z)2 dw dx dy dz\displaystyle \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \int \limits_0^\infty \frac{1}{wxyz \left(w+x+y+z+\dfrac1w + \dfrac1x + \dfrac1y + \dfrac1z \right)^2} \ dw \ dx \ dy \ dz

If the integral above is equal to ABζ(C)\dfrac{A}{B} \zeta(C), where AA and BB are coprime positive integers and CC is an integer, find A+B+CA+B+C.

Notation: ζ()\zeta(\cdot) denotes the Riemann zeta function.

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