1. The problem is to evaluate the integral $$I = \int \tan(\tan t) \cdot t \, dt.$$\n\n2. This integral is quite complex because it involves a composition of the tangent function inside another tangent function multiplied by $t$. There is no standard elementary antiderivative for $\tan(\tan t)$.\n\n3. Generally, integrals of compositions like $\tan(\tan t)$ do not have closed-form solutions in elementary functions.\n\n4. To solve or approximate this integral, one might consider numerical methods or series expansions, but no simple formula applies.\n\n5. Therefore, the integral $$I = \int \tan(\tan t) \cdot t \, dt$$ cannot be expressed in elementary closed form.