"Feel the bern" integral

Calculus Level 5

\[ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \dfrac{db \, de_{1} \, dr \, dn \, di \, de_{2} }{1- be_{1}rnie_{2} } \]

If the above thing can be expressed in the form \( \dfrac{a \pi^{b}}{c} \), with \(a, c\) as coprime positive integers, find \(a+b+c\).

Details and assumptions:

There's no \(e \approx 2.71828... \) or \( i = \sqrt{-1} \). The \( b, e_{1}, r, n, i, e_{2} \) are all variables.

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