Simple technique to find $\int \frac{d x}{\cos x-\csc x}$

Question

$$\begin{aligned}\int \frac{d x}{\cos x-\csc x} =& \int \frac{\sin x}{\sin x \cos x-1} d x \\ =& \int \frac{(\cos x+\sin x)-(\cos x-\sin x)}{2 \sin x \cos x-2} d x \\ =& \int \frac{d(\cos x-\sin x)}{1+(\cos x-\sin x)^{2}}-\int \frac{d(\sin x+\cos x)}{(\sin x+\cos x)^{2}-3} \\ =&\arctan(\cos x-\sin x)+\frac{1}{2 \sqrt{3}} \ln \left|\frac{\sin x+\cos x+\sqrt{3}}{\sin x+\cos x-\sqrt3}\right|+C\end{aligned}$$