  An $arctan$-electrodynamics coupled with the gravitational field is
investigated. We obtain the regular black hole solution that at $r\rightarrow
\infty$ gives corrections to the Reissner-Nordstr\"om solution. The corrections
to Coulomb's law at $r\rightarrow\infty$ are found. We evaluate the mass of the
black hole that is a function of the dimensional parameter $\beta$ introduced
in the model. The magnetically charged black hole was investigated and we have
obtained the magnetic mass of the black hole and the metric function at
$r\rightarrow \infty$. The regular black hole solution is obtained at
$r\rightarrow 0$ with the de Sitter core. We show that there is no singularity
of the Ricci scalar for electrically and magnetically charged black holes.
Restrictions on the electric and magnetic fields are found that follow from the
requirement of the absence of superluminal sound speed and the requirement of a
classical stability.
