\(\int \frac{\operatorname{cosec} x d x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)}=\)

Let \(\mathrm{I}=\int \frac{\operatorname{cosec} x \mathrm{~d} x}{\cos ^2\left(1+\log \tan \frac{x}{2}\right)} \mathrm{d} x\)
Let \(1+\log \left(\tan \frac{x}{2}\right)=\mathrm{t}\)
Differentiating both sides w.r.t. t, we get \(\frac{1}{\tan \frac{x}{2}} \sec ^2 \frac{x}{2} \times \frac{1}{2} \mathrm{~d} x=\mathrm{dt}\)
\(\begin{aligned}
& \therefore \quad \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \mathrm{~d} x=\mathrm{dt} \\
& \therefore \quad \operatorname{cosec} x \mathrm{~d} x=\mathrm{dt} \\
& \therefore \quad I=\int \frac{1}{\cos ^2 t} d t \\
& =\int \sec ^2 t d t \\
& =\tan (\mathrm{t})+\mathrm{c} \\
& =\tan \left(1+\log \left(\tan \frac{x}{2}\right)\right)+\mathrm{c}
\end{aligned}\)