\[\int_0^1 {\rm Li}_2 \left(\dfrac{x}{2}\right)\log(1-x)\; dx = A+\dfrac{\log^B C}{D}+\dfrac{\zeta(E)\log F}{G}-\zeta(H)-\zeta(I)-J\log(K)\]

If the above equation holds true for positive integers \(A,B,C,D,E,F,G,H,I,J,K\) not necessarily distinct then find \(A+B+C+D+E+F+G+H+I+J+K\).

**Notations:**

- \({\rm Li}_s(\cdot)\) denotes the polylogarithm function.
- \(\zeta(\cdot)\) denotes the Riemann zeta function.

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