The total cross section and differential cross sections for the production of $B_c^{\pm }$ mesons, times their branching fraction to $J/\psi {\pi }^{\pm }$, are measured relative to those for the production of $B^{\pm }$ mesons, times their branching fraction to $J/\psi K^{\pm }$. The data used for this study correspond to an integrated luminosity of $20.3{\mathrm{fb}}^{-1}$ of $pp$ collisions recorded by the ATLAS detector at the Large Hadron Collider in 2012 at a center-of-mass energy of $\sqrt{s}=8\mathrm{TeV}$. The measurement is performed differentially in bins of transverse momentum $p_T$ for $13\mathrm{GeV}<p_T\left(B_c^{\pm }\right)<22\mathrm{GeV}$ and $p_T\left(B_c^{\pm }\right)>22\mathrm{GeV}$ and in bins of rapidity $y$ for $\left|y\right|<0.75$ and $0.75<\left|y\right|<2.3$. The relative cross section times branching fraction for the full range $p_T>13\mathrm{GeV}$ and $\left|y\right|<2.3$ is $(0.34\pm 0.04_{\mathrm{stat}}{{}_{-0.02}^{+0.06}}_{\mathrm{sys}}\pm 0.01_{\text{lifetime}})\%$. The differential measurements suggest that the production cross section of the $B_c^{\pm }$ decreases faster with $p_T$ than the production cross section of the $B^{\pm }$, while no significant dependence on rapidity is observed.