Can You Believe A Closed Form Exists?

Calculus Level 5

\[\begin{eqnarray}&&\int _{ 0 }^{ 1 }{ \left[ \ln { (\Gamma (x)) } \right] ^{ 2 } \, dx } \\ &&=\dfrac { \gamma ^{ A } }{ B } +\dfrac { \gamma }{ C } \ln { \sqrt { D\pi } } +\dfrac { \pi ^{ E } }{ F } +\dfrac { G }{ H } \ln { ^{ I } } { \sqrt { J\pi } }+\dfrac { \zeta ^{ \prime \prime }(K) }{ L{ \pi }^{ M } } -\dfrac { N\gamma \zeta ^{ \prime }(O) }{ \pi ^{ P } } -\dfrac { \ln { (Q\pi) } \zeta ^{ \prime }(R) }{ { \pi }^{ S } }\end{eqnarray} \]

If the equation above holds true for positive integers \(A,B,C,\ldots,Q,R,S\), then find the minimum value of \(A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R+S\).

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