  We consider the three-dimensional Hamiltonian for Bi$_2$Se$_3$, a
second-generation topological insulator, under the effect of a periodic drive
for both in-plane and out-of-plane fields. As it will be shown by means of
high-frequency expansions up to second order in the Floquet Hamiltonian, the
driving induces anisotropies in the Dirac cone and opens up a quasienergy gap
for in-plane elliptically polarized fields. Analytic expressions are obtained
for the renormalized velocities and the quasienergy gap. These expressions are
then compared to numerical calculations performed by discretizing the
Hamiltonian in a one-dimensional lattice and following a staggered fermion
approach, achieving a remarkable agreement. We believe our work may have an
impact on the transport properties of topological insulators.
