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

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
\(\frac{1}{\tan \frac{x}{2}} \sec ^2 \frac{x}{2} \times \frac{1}{2} \mathrm{~d} x=\mathrm{dt} \)
\( \therefore \frac{1}{2 \sin \frac{x}{2} \cos \frac{x}{2}} \mathrm{~d} x=\mathrm{dt} \)
\( \therefore \operatorname{cosec} x \mathrm{~d} x=\mathrm{dt} \)
\( \therefore \mathrm{I}=\int \frac{1}{\cos ^2 \mathrm{t}} \mathrm{dt}=\int \sec ^2 \mathrm{t} d \mathrm{t} \)
\( =\tan (\mathrm{t})+\mathrm{c} \)
\( =\tan \left(1+\log \left(\tan \frac{x}{2}\right)\right)+c\)