Summing up all digamma roots

Calculus Level 5

\[\Large \sum_{x \in \Psi} \frac{1}{x(x-2)}=\dfrac{1-\frac{\gamma^A}{B}-\frac{\pi^C}{D}}{1-\gamma}\]

Let \(\Psi\) is the set consisting of all roots of the digamma function.

Given that \(A,B,C\) and \(D\) are positive integers satisfying the equation above, find \(A+B+C+D\).

Notation: \(\gamma\) denotes the Euler-Mascheroni constant, \(\displaystyle\gamma = \lim_{n\to\infty} \left( - \ln(n) + \sum_{k=1}^n \dfrac1k \right) \approx 0.5772 \).

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