Hot Integral - 29

Calculus Level 5

\[\large \int_{0}^{1} \int_{0}^{1} \left( \frac{(x-1)(y-1)(xy-1)}{(x+1)(y+1)(xy+1)} \right)^2\frac{1}{(x+xy+1)(y+xy+1)} \text{d}x \text{d}y \]

The above integral can be expressed as:

\[Ai\text{Li}_B\left( \frac{C}{D} - \frac{\sqrt{E}}{F}i\right)\sqrt{G} - H\sqrt{I}i\text{Li}_J\left( \frac{K}{L} - \frac{\sqrt{M}}{N}i\right) + O + P\pi^Q \]

Solve \( A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+128 \)

where \(i=\sqrt{-1}\) and \( \gcd(C,D)=\gcd(K,L)=1 \)

All capital alphabets are positive integers.

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