  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.
