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We investigate how the electron-vibron coupling influences electron transport via an anisotropic magnetic molecule, such as a single-molecule magnet (SMM) Fe sub(4), by using a model Hamiltonian with parameter values obtained from density-functional theory (DFT). The magnetic anisotropy parameters, vibrational energies, and electron-vibron coupling strengths of the Fe sub(4) are computed using DFT. In sequential electron tunneling, we find that the magnetic anisotropy gives rise to new features in the conductance peaks arising from vibrational excitations. The magnetic anisotropy also introduces vibrational satellite peaks whose position and height are modified with the direction and magnitude of applied B field. The new features appear because the magnetic anisotropy barrier is of the same order of magnitude as the energies of vibrational modes with significant electron-vibron coupling. Our findings clearly show the interesting interplay between magnetic anisotropy and electron-vibron coupling in electron transport via the Fe sub(4). Similar behavior can be observed in transport via other anisotropic magnetic molecules.