Step 1: Substitution. Let \( u = 1 + \log \left( \tan \frac{x}{2} \right) \). Differentiating yields \[\frac{du}{dx} = \frac{1}{\tan \frac{x}{2}} \cdot \sec^2 \frac{x}{2} \cdot \frac{1}{2}.\] Thus, \[du = \frac{\csc x}{2} \, dx \quad \Rightarrow \quad 2 du = \frac{\csc x}{\cos^2 u} \, dx.\]Step 2: Substituting into the integral. The integral transforms to: \[I = 2 \int \sec^2 u \, du.\]Step 3: Solving the integral. The integral of \( \sec^2 u \) is \( \tan u \): \[I = 2 \tan u + C.\]Step 4: Substitute back the value of \( u \). Replacing \( u \) with \( 1 + \log \left( \tan \frac{x}{2} \right) \): \[I = 2 \tan \left( 1 + \log \left( \tan \frac{x}{2} \right) \right) + C.\]The constant factor of 2 can be simplified, yielding the final result: \[I = \tan \left( 1 + \log \left( \tan \frac{x}{2} \right) \right) + C.\]