$\int \frac{e^{\cot x}}{{\sin }^{2}x}\left(2\mathrm{lncos}ecx+\sin 2x\right)dx$

$\begin{array}{l}t=\cot x\Rightarrow dt=-{\mathrm{cosec}}^{2}xdx\\ I=\int \frac{e^{\cot x}}{{\sin }^{2}x}\left(\ln \cos e{c}^{2}x+\frac{2\tan x}{1+{\tan }^{2}x}\right)dx\end{array}$

$\begin{array}{l}=-\int e^t\left(\ln \left(1+t^2\right)+\frac{\frac{2}{t}}{1+\frac{1}{t^2}}\right)dt\\ =-e^t\ln \left|1+t^2\right|+c=-2e^{\cot x}\ln |\cos \mathrm{ec}x|+c\\ =2e^{\cot x }\ln \left|\sin x\right|+c\end{array}$