Answer
\[-\ln |1+\cot x|+C\]
Where $C$ is constant of integration
Work Step by Step
Let \[I=\int \frac{\csc^2 x}{1+\cot x}\]
Put \[t=1+\cot x\;\;...(1)\;\;\Rightarrow dt=-\csc^2 x\:dx\]
\[\Rightarrow I=-\int \frac{dt}{t}\]
\[\Rightarrow I=-\ln |t|+C\]
Where $C$ is constant of integration
Using (1)
\[\Rightarrow I=-\ln |1+\cot x|+C\]
Hence,
\[\int \frac{\csc^2 x}{1+\cot x}=-\ln |1+\cot x|+C\]
