  We study the semileptonic differential decay rates of $B_c$ meson to S-wave
charmonia, $\eta_c$ and $J/\Psi$, at the next-to-leading order accuracy in the
framework of NRQCD. In the heavy quark limit, $m_b \to \infty$, we obtain
analytically the asymptotic expression for the ratio of NLO form factor to LO
form factor. Numerical results show that the convergence of the ratio is
perfect. At the maximum recoil region, we analyze the differential decay rates
in detail with various input parameters and polarizations of $J/\psi$, which
can now be checked in the LHCb experiment. Phenomenologically, the form factors
are extrapolated to the minimal recoil region, and then the $B_c$ to charmonium
semileptonic decay rates are estimated.
