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In this article, we study the three-pion $B_c$-decays $B_c^+ \to J/\psi\,\pi^+\pi^-\pi^+$, $\eta_c\, \pi^+\pi^-\pi^+ $ with dominance of theintermediate axial-vector meson $a_1(1260)$ and vector meson $\rho(770)$ in the$\pi^+\pi^-\pi^+$ and $\pi^+\pi^-$ invariant mass distributions respectively,and make predictions for the branching fractions and differential decay widths.The ratio between the decays $B_c^+ \to J/\psi\, \pi^+\pi^-\pi^+$ and $B_c^+\to J/\psi\, \pi^+ $ is compatible with the experimental data withinuncertainties, other predictions can be confronted with the experimental datain the future at the LHCb.

The $B_c$-decays $B_c^+ \to J/\psi \pi^+\pi^-\pi^+$, $\eta_c \pi^+\pi^-\pi^+ $