Abstract In this study, we investigate the behavior of free inertial active Brownian particles in the presence of thermal noise. While finding a closed-form solution for the joint distribution of positions, orientations, and velocities using the Fokker–Planck equation is generally challenging, we utilize a Laplace transform method to obtain the exact temporal evolution of all dynamical moments in arbitrary dimensions. Our expressions in d dimensions reveal that inertia significantly impacts steady-state kinetic temperature and swim pressure while leaving the late-time diffusivity unchanged. Notably, as a function of activity and inertia, the steady-state velocity distribution exhibits a remarkable re-entrant crossover from ‘passive’ Gaussian to ‘active’ non-Gaussian behaviors. We construct a corresponding ‘phase diagram’ using the exact expression of the d -dimensional kurtosis. Our analytic expressions describe steady states and offer insights into time-dependent crossovers observed in moments of velocity and displacement. Our calculations can be extended to predict up to second-order moments for run-and-tumble particles and the active Ornstein–Uhlenbeck process (AOUP). Additionally, the kurtosis shows differences from AOUP.