$$ \displaystyle\int \dfrac{x}{{\left(\sin\left(x\right)\right)}^{5}}\, \mathrm d x = -{{9\,x\,\sin \left(8\,x\right)\,\ln \left(\sin ^2x+\cos ^2x+2\, \cos x+1\right)-9\,i\,x\,\cos \left(8\,x\right)\,\ln \left(\sin ^2x +\cos ^2x+2\,\cos x+1\right)-36\,x\,\sin \left(6\,x\right)\,\ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1\right)+36\,i\,x\,\cos \left(6\,x \right)\,\ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1\right)+54\,x\, \sin \left(4\,x\right)\,\ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1 \right)-54\,i\,x\,\cos \left(4\,x\right)\,\ln \left(\sin ^2x+\cos ^ 2x+2\,\cos x+1\right)-36\,x\,\sin \left(2\,x\right)\,\ln \left( \sin ^2x+\cos ^2x+2\,\cos x+1\right)+36\,i\,x\,\cos \left(2\,x \right)\,\ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1\right)-9\,i\,x\, \ln \left(\sin ^2x+\cos ^2x+2\,\cos x+1\right)-9\,x\,\sin \left(8\, x\right)\,\ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)+9\,i\,x\, \cos \left(8\,x\right)\,\ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1 \right)+36\,x\,\sin \left(6\,x\right)\,\ln \left(\sin ^2x+\cos ^2x- 2\,\cos x+1\right)-36\,i\,x\,\cos \left(6\,x\right)\,\ln \left( \sin ^2x+\cos ^2x-2\,\cos x+1\right)-54\,x\,\sin \left(4\,x\right)\, \ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)+54\,i\,x\,\cos \left(4\,x\right)\,\ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)+ 36\,x\,\sin \left(2\,x\right)\,\ln \left(\sin ^2x+\cos ^2x-2\,\cos x+1\right)-36\,i\,x\,\cos \left(2\,x\right)\,\ln \left(\sin ^2x+ \cos ^2x-2\,\cos x+1\right)+9\,i\,x\,\ln \left(\sin ^2x+\cos ^2x-2 \,\cos x+1\right)+18\,i\,x\,\sin \left(8\,x\right)\,{\rm atan2} \left(\sin x , \cos x+1\right)+18\,x\,\cos \left(8\,x\right)\, {\rm atan2}\left(\sin x , \cos x+1\right)-72\,i\,x\,\sin \left(6\,x \right)\,{\rm atan2}\left(\sin x , \cos x+1\right)-72\,x\,\cos \left(6\,x\right)\,{\rm atan2}\left(\sin x , \cos x+1\right)+108\,i \,x\,\sin \left(4\,x\right)\,{\rm atan2}\left(\sin x , \cos x+1 \right)+108\,x\,\cos \left(4\,x\right)\,{\rm atan2}\left(\sin x , \cos x+1\right)-72\,i\,x\,\sin \left(2\,x\right)\,{\rm atan2}\left( \sin x , \cos x+1\right)-72\,x\,\cos \left(2\,x\right)\,{\rm atan2} \left(\sin x , \cos x+1\right)+18\,x\,{\rm atan2}\left(\sin x , \cos x+1\right)+18\,i\,x\,\sin \left(8\,x\right)\,{\rm atan2}\left( \sin x , 1-\cos x\right)+18\,x\,\cos \left(8\,x\right)\,{\rm atan2} \left(\sin x , 1-\cos x\right)-72\,i\,x\,\sin \left(6\,x\right)\, {\rm atan2}\left(\sin x , 1-\cos x\right)-72\,x\,\cos \left(6\,x \right)\,{\rm atan2}\left(\sin x , 1-\cos x\right)+108\,i\,x\,\sin \left(4\,x\right)\,{\rm atan2}\left(\sin x , 1-\cos x\right)+108\,x \,\cos \left(4\,x\right)\,{\rm atan2}\left(\sin x , 1-\cos x\right)- 72\,i\,x\,\sin \left(2\,x\right)\,{\rm atan2}\left(\sin x , 1-\cos x \right)-72\,x\,\cos \left(2\,x\right)\,{\rm atan2}\left(\sin x , 1- \cos x\right)+18\,x\,{\rm atan2}\left(\sin x , 1-\cos x\right)+18\,i \,\sin \left(8\,x\right)\,{\it li}_{2}(e^{i\,x})+18\,\cos \left(8\,x \right)\,{\it li}_{2}(e^{i\,x})-72\,i\,\sin \left(6\,x\right)\, {\it li}_{2}(e^{i\,x})-72\,\cos \left(6\,x\right)\,{\it li}_{2}(e^{i \,x})+108\,i\,\sin \left(4\,x\right)\,{\it li}_{2}(e^{i\,x})+108\, \cos \left(4\,x\right)\,{\it li}_{2}(e^{i\,x})-72\,i\,\sin \left(2\, x\right)\,{\it li}_{2}(e^{i\,x})-72\,\cos \left(2\,x\right)\, {\it li}_{2}(e^{i\,x})+18\,{\it li}_{2}(e^{i\,x})-18\,i\,\sin \left( 8\,x\right)\,{\it li}_{2}(-e^{i\,x})-18\,\cos \left(8\,x\right)\, {\it li}_{2}(-e^{i\,x})+72\,i\,\sin \left(6\,x\right)\,{\it li}_{2}( -e^{i\,x})+72\,\cos \left(6\,x\right)\,{\it li}_{2}(-e^{i\,x})-108\, i\,\sin \left(4\,x\right)\,{\it li}_{2}(-e^{i\,x})-108\,\cos \left(4 \,x\right)\,{\it li}_{2}(-e^{i\,x})+72\,i\,\sin \left(2\,x\right)\, {\it li}_{2}(-e^{i\,x})+72\,\cos \left(2\,x\right)\,{\it li}_{2}(-e ^{i\,x})-18\,{\it li}_{2}(-e^{i\,x})-36\,x\,\sin \left(7\,x\right)+ 36\,i\,\sin \left(7\,x\right)+36\,i\,x\,\cos \left(7\,x\right)+36\, \cos \left(7\,x\right)+132\,x\,\sin \left(5\,x\right)-140\,i\,\sin \left(5\,x\right)-132\,i\,x\,\cos \left(5\,x\right)-140\,\cos \left( 5\,x\right)+132\,x\,\sin \left(3\,x\right)+140\,i\,\sin \left(3\,x \right)-132\,i\,x\,\cos \left(3\,x\right)+140\,\cos \left(3\,x \right)-36\,x\,\sin x-36\,i\,\sin x+36\,i\,x\,\cos x-36\,\cos x }\over{48\,\left(\sin \left(8\,x\right)-i\,\cos \left(8\,x\right)-4 \,\sin \left(6\,x\right)+4\,i\,\cos \left(6\,x\right)+6\,\sin \left( 4\,x\right)-6\,i\,\cos \left(4\,x\right)-4\,\sin \left(2\,x\right)+4 \,i\,\cos \left(2\,x\right)-i\right)}} $$

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