\[\displaystyle \int\limits_{0}^{\pi} t^3\log^8(2\sin t) \, dt\]

If the above integral can be expressed as : \[\frac{A\pi^C}{B}+\frac{D\pi^F\zeta(G)^H}{E}+I\pi^J\zeta(K)^L + M\pi^N\zeta(O)\zeta(P)+\frac{Q}{U}\pi^R\zeta(S)\zeta(T)\]

Evaluate \(A+B+C+D+E+F+G+H+I+J+K+L+M+N+O+P+Q+R+S+T+U\)

**Details and Assumptions**

\(A,B,...,S,T,U\) all are integers.

\(\gcd(A,B)=\gcd(D,E)=\gcd(Q,U)=1\)

This is a part of Hot Integrals

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