Who's up to the challenge? 19

Calculus Level 5

\[ \large \int_0^1 ( \ln x)^3 \ln (1-x^2) \, dx \]

The value the integral above is equal to

\[ -\dfrac { A }{ B } \zeta (C)-\dfrac { D }{ F } { \pi }^{ G }+H-\dfrac { \pi ^{ J } }{ K } -L\ln { 2 }, \]

where \(A,B,C,D,F,G,H,J,K\) and \(L \) are positive integers with \(\gcd(A,B) = \gcd(D,F) = 1 \).

Find \(A+B+C+D+F+G+H+J+K+L\).


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