  The semileptonic decays $B_c^+\rightarrow P(V) +\ell^++\bar{\nu}_\ell$ and
the nonleptonic decays $B_c^+\rightarrow P(V)+L$, where $P(V)$ denotes a
pseudoscalar (vector) charmonium or ($\bar{b}s$)-meson, and $L$ denotes a light
meson, are studied in the framework of improved instantaneous Bethe-Salpeter
(BS) equation and the Mandelstam formula. The numerical results (width and
branching ratio of the decays) are presented in tables, and in order to compare
conveniently, those obtained by other approaches are also put in the relevant
tables. Based on the fact that the ratio
$\frac{\mathcal{BR}(B_c^+\rightarrow\psi(2S)\pi^+)}{\mathcal{BR}(B_c^+\rightarrow
J/\psi \pi^+)}=0.24^{+0.023}_{-0.040}$ estimated here is in good agreement with
the observation by the LHCb $\frac{\mathcal{BR}(B_c^+\rightarrow
\psi(2S)\pi^+)}{\mathcal{BR}(B_c^+\rightarrow J/\psi
\pi^+)}=0.250\pm0.068(\mathrm{stat})\pm0.014(\mathrm{syst})\pm0.006(\mathcal{B})$,
one may conclude that with respect to the decays the present framework works
quite well.
