  In this paper, we study rare decays $B_c\rightarrow D^{(*)}_{s,d}l\bar{l}$
within the Standard Model. The penguin, box, annihilation, color-favored
cascade and color-suppressed cascade contributions are included. Based on our
calculation, the annihilation and color-favored cascade diagrams play important
roles in the differential branching fractions, forward-backward asymmetries,
longitudinal polarizations of the final vector mesons and leptonic longitudinal
polarization asymmetries. More importantly, color-favored cascade decays
largely enhance the resonance cascade contributions. To avoid the resonance
cascade contribution pollution, new cutting regions are put forward.
