The total cross section and differential cross sections for the production of $Bc\pm$ 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 $s=8$ TeV. The measurement is performed differentially in bins of transverse momentum $pT$ for $13$ GeV GeV and $pT(Bc\pm )>22$ GeV and in bins of rapidity $y$ for and . The relative cross section times branching fraction for the full range $pT>13$ GeV and is $(0.34\pm 0.04stat-0.02+0.06sys\pm 0.01lifetime)\%$. The differential measurements suggest that the production cross section of the $Bc\pm$ decreases faster with $pT$ than the production cross section of the $B\pm$, while no significant dependence on rapidity is observed.