$$ \displaystyle\int x{\cdot}{\left(\sin\left(x\right)\right)}^{-1}\, \mathrm d x = -{{x\,\ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1\right)-x\,\ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)+2\,i\,x\,{\rm atan2} \left(\sin x , \cos x+1\right)+2\,i\,x\,{\rm atan2}\left(\sin x , 1- \cos x\right)+2\,i\,{\it li}_{2}(e^{i\,x})-2\,i\,{\it li}_{2}(-e^{i \,x})}\over{2}} $$

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