Let \[\displaystyle f(t)=\int\limits_{0}^{\pi} \sin\left(t\sin\left(\frac{\pi}{4}\right)\sin x\right) \sinh\left(t\sin\left(\frac{\pi}{4}\right)\sin x\right) \, dx\]

Then, \[\mathcal{L}\left\{f(t)\right\}(s) = \frac{A\pi^K}{(s^B + C)^{D/E}}\sin\left(\frac{F}{G}\tan^{-1}\left(\frac{H}{s^J}\right)\right)\]

Evaluate \(A+B+C+D+E+F+G+H+J+K\)

**Details and Assumptions:**

- \(\mathcal{L}\) represents Laplace transform.
- \(D,E\) and \(F,G\) are co-prime.

This is a part of Hot Integrals

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