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In this paper we study the semileptonic decays of $B_c^+\toD^{(*)}_{(s)}(l^+\nu_l,l^+l^-,\nu\bar\nu)$ (here $l$ stands for $e$, $\mu$, or$\tau$). After evaluating the $B_c^+ \to (D_{(s)},D^*_{(s)})$ transition formfactors $F_{0,+,T}(q^2)$ and $V(q^2), A_{0,1,2}(q^2), T_{1,2,3}(q^2)$ byemploying the perturbative QCD factorization approach, we calculate thebranching ratios for all these semileptonic decays. Our predictions for thevalues of the $B_c^+ \to D_{(s)}$ and $B_c^+ \to D^*_{(s)}$ transition formfactors are consistent with those obtained by using other methods. Thebranching ratios of the decay modes with $\bar\nu\nu$ are almost an order ofmagnitude larger than the corresponding decays with $l^+l^-$ after thesummation over the three neutrino generations. The branching ratios for thedecays with $b\to d$ transitions are much smaller than those decays with the$b\to s$ transitions, due to the Cabibbo-Kobayashi-Maskawa suppression. Wedefine ratios $R_D$ and $R_{D^*}$ for the branching ratios with the $\tau$lepton versus $\mu$, $e$ lepton final states to cancel the uncertainties of theform factors, which could possibly be tested in the near future.

Semileptonic decays $B_c^+\to D^{(*)}_{(s)}(l^+\nu,l^+l^-,\nu\bar\nu)$ in the perturbative QCD approach