Answer
$$\int t \csc t \cot t d t =-t\csc t+\ln|\csc t+\cot t|+c$$
Work Step by Step
Since $$
\int t \csc t \cot t d t
$$
Integrate by parts, let
\begin{align*}
u&=t\ \ \ \ \ \ \ dv=\csc t \cot t d t\\
du&=dt\ \ \ \ \ \ \ v=-\csc t
\end{align*}
Then
\begin{align*}
\int t \csc t \cot t d t&=uv-\int vdu\\
&=-t\csc t+\int \csc t dt\\
&=-t\csc t+\ln|\csc t+\cot t|+c
\end{align*}
