\[\displaystyle \int\limits_{-\infty}^{\infty} \frac{x^4+x^2+x}{\sinh x - i} dx = \frac{\pi}{A}+i(\frac{\pi^B}{C}+\frac{D\pi^E}{F})\]

Calculate \(A+B+C+D+E+F\)

**Details And Assumptions**

\(\gcd(D,E)=1\)

\(A,B,C,D,E,F\) are integers.

\(i=\sqrt{-1}\)

This is a part of Hot Intergals

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