[tex]\displaystyle\int\cot^3x\,\mathrm dx=\int\cot^2x\cot x\,\mathrm dx=\int(\csc^2x-1)\cot x\,\mathrm dx[/tex]
[tex]\displaystyle=\int\csc^2x\cot x\,\mathrm dx-\int\cot x\,\mathrm dx[/tex]
For the first integral, substitute [tex]y=\cot x[/tex] so that [tex]\mathrm dy=-\csc^2x\,\mathrm dx[/tex]. This leaves you with two standard integrals,
[tex]=\displaystyle-\int y\,\mathrm dy-\int\cot x\,\mathrm dx[/tex]
[tex]=-\dfrac12y^2+\ln|\cot x+\csc x|+C[/tex]
[tex]=-\dfrac12\cot^2x+\ln|\cot x+\csc x|+C[/tex]
