  We present a combined analytical and numerical micromagnetic study of the equilibrium energy, size and shape of anti-skyrmionic magnetic configurations. Anti-skyrmions can be stabilized when the Dzyaloshinskii-Moriya interaction has opposite signs along two orthogonal in-plane directions, breaking the magnetic circular symmetry. We compare the equilibrium energy, size and shape of anti-skyrmions and skyrmions that are stabilized respectively in environments with anisotropic and isotropic Dzyaloshinskii-Moriya interaction, but with the same strength of the magnetic interactions.When the dipolar interactions are neglected the skyrmion and the anti-skyrmion have the same energy, shape and size in their respective environment. However, when dipolar interactions are considered, the energy of the anti-skyrmion is strongly reduced and its equilibrium size increased with respect to the skyrmion. While the skyrmion configuration shows homochiral N\'{e}el magnetization rotations, anti-skyrmions show partly N\'{e}el and partly Bloch rotations. The latter do not produce magnetic charges and thus cost less dipolar energy. Both magnetic configurations are stable when the magnetic energies almost cancel each other, which means that a small variation of one parameter can drastically change their configuration, size and energy. 
Micromagnetics of anti-skyrmions in ultrathin films