Hot Integral - 13

$\displaystyle \int\limits_{0}^{1} \frac{t^{n-1}\log^2 t}{f(t)} dt= \frac{K\gamma + n(An\psi^{(X)}(n) + B\psi^{(Y)}(n) + Cn\zeta(D) + \pi^E) + P\psi^{(Z)}(n+Q)}{n^L}$

where , $\displaystyle f(x) = \lim_{\xi \to 0} \frac{\xi^2}{\; _2F_1(\xi,\xi;1;x) - 1}$

Calculate $A+B+C+D+E+K+L+P+Q+X+Y+Z$

Clarifications:

• $A,B,C,D,E,K,L,P,Q,X,Y,Z$ are integers.

• $\psi^{(X)}$ etc. represent their usual meanings (i.e. Polygamma function).

• $n \in N$ where $N$ is the set of natural numbers.

• $\gamma$ is Euler-Mascheroni constant

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This is a part of Hot Integrals