How would you solve it part-3

Calculus Level 5

$\large \displaystyle \int_{0}^{\pi} \tan^{-1}\left(\dfrac{\ln(\sin(x))}{x}\right)\mathbb{d}x = -\pi^{\alpha}\tan^{-1}\left(\dfrac{\beta\ln^{\eta}{(\gamma)}}{\pi^{\delta}}\right)$ If the above integral equals the above closed form for positive integers $$\alpha,\beta,\gamma,\delta,\eta$$ and $$\gamma$$ is a prime, then find $\alpha +\beta^{\gamma} +\delta +\eta$ Try part 1 and part 2

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