$I=\int \cot [\dfrac{\pi}{2} -(x+\dfrac{\pi}{3})] \cot (x-\dfrac{\pi}{6})dx=-\int \cot^2 (x-\dfrac{\pi}{6})dx=-\int \cot^2 (x-\dfrac{\pi}{6})d(x-\dfrac{\pi}{6})$
$=-\int \cot^2 t dt =-\int \dfrac{\cos^2 t}{\sin^2 t }dt=-\int \dfrac{1-\sin^2 t}{\sin^2 t}dt=-\int \dfrac{1}{\sin^2 t}dt +\int dt$
$=\cot t + t + C$