The deterministic Landau-Lifshitz-Gilbert equation has been used to investigate the nonlinear dynamics of magnetization and the specific power loss in magnetic nanoparticles with uniaxial anisotropy driven by a rotating magnetic field, generalizing the results obtained for the isotropic case found by P. F. de Châtel, I. Nándori, J. Hakl, S. Mészáros, and K. Vad [J. Phys. Condens. Matter 21, 124202 (2009)]. As opposed to many applications of magnetization reversal in single-domain ferromagnetic particles, where losses must be minimized, in this paper, we study the mechanisms of dissipation used in cancer therapy by hyperthermia, which requires the enhancement of energy losses. We show that for circularly polarized field, the energy loss per cycle is decreased by the anisotropy compared to the isotropic case when only dynamical effects are taken into account. Thus, in this case, in the low-frequency limit, a better heating efficiency can be achieved for isotropic nanoparticles. The possible role of thermal fluctuations is also discussed. Results obtained are compared to experimental data.