Hot Integral - 13

Calculus Level 5

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

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

This is a part of Hot Integrals