Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.3 Exercises - Page 530: 51

$$\int \frac{\cot ^{2} t}{\csc t} d t =\ln \left|\csc t+\cot t\right|+\cos \left(t\right)+C$$

$$ \int \frac{\cot ^{2} t}{\csc t} d t $$ Since \begin{align*} \int \frac{\cot ^{2} t}{\csc t} d t&= \int \frac{\csc ^{2} t-1}{\csc t} d t\\ &=\int (\csc t-\sin t)dt\\ &=\ln \left|\csc t+\cot t\right|+\cos \left(t\right)+C \end{align*}