Summary: Let \(\mathscr{H}\) be a real Hilbert space, and \(f:\mathscr{H} \to \mathbb{R}\) be a convex twice differentiable function whose solution set \(\operatorname{argmin}f\) is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping \[ \ddot{x}(t) + \alpha \dot{x}(t) + \delta\nabla^2 f (x(t))\dot{x}(t) + \beta(t)\nabla f (x(t)) + cx(t) = 0 \] where \(\alpha\), \(c\), \(\delta\) are three positive constants, and the time scale parameter \(\beta\) is a positive nondecreasing function such that \(\lim_{t\to +\infty} \beta(t) = +\infty\). Under some assumptions on the parameter \(\beta\), we will show rapid convergence of values, strong convergence toward the minimum norm element of \(\operatorname{argmin}f\), and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter \(\beta\) improves the rate of convergences mentioned above. As particular cases of \(\beta\), we set \(\beta(t) = t^p \ln^q(t)\), for \((p, q) \in (\mathbb{R}_+)^2 \backslash \{(0, 0)\}\), and \(\beta(t) = e^{\gamma t^p}\), for \(p \in ]0, 1[\) and \(\gamma > 0\). The manuscript concludes with two numerical examples and comments on their performance.