Hot Integral - 23

$\displaystyle \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \int\limits_{0}^{1} \left(\prod_{1\leq j <k\leq n} \frac{u_k-u_j}{u_k+u_j}\right)^2 dt_1 dt_2 dt_3 dt_4 dt_5$

If the above integral can be expressed as :

$A\log^B C + \log^D E(F\pi^G+H)+\log I\left(\frac{J}{L}\pi^K-M\zeta(N)-O\right)+\pi^P\left(\frac{Q}{R}\pi^S-\frac{T}{U}\right)-V\zeta(W)-X\text{Li}_Y\left(\frac{Z_1}{Z_2}\right)+Z_3$

Evaluate $A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R+S+T+U+V+W+X+Y+Z_1+Z_2+Z_3$

Details and Assumptions

• $A,B,C,...,X,Y,Z_1,Z_2,Z_3$ are all positive integers.

• $\gcd(I,L)=\gcd(Q,R)=\gcd(T,U)=\gcd(Z_1,Z_2)=1$

• $u_k=\prod_{i=1}^{k}t_i$

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This is a part of Hot Integrals