The value of the integral $\int_{0}^{3α}cosec(x-α)cosec(x-2α)dx$ is

$2 cosec\, α \log\left(\frac{1}{2}\sec α\right)$

$I=\frac{1}{\sin α}\int_{0}^{3α}\frac{\sin α\,dt}{\sin(x-α)\sin(x-2α)}$

$=\frac{1}{\sin α}\int_{0}^{3α}\frac{\sin\{(x-α)-(x-2α)\}dx}{\sin(x-α)\sin(x-2α)}$

$=\frac{1}{\sin α}\int_{0}^{3α}\cot(x-2α)-\cot(x-α)dx$

$=2 cosec\, α \log\left(\frac{1}{2}\sec α\right)$