  In this paper we analyze the nuclear fusion rate between equal nuclei for all
five different nuclear burning regimes in dense matter (two thermonuclear
regimes, two pycnonuclear ones, and the intermediate regime). The rate is
determined by Coulomb barrier penetration in dense environments and by the
astrophysical S-factor at low energies. We evaluate previous studies of the
Coulomb barrier problem and propose a simple phenomenological formula for the
reaction rate which covers all cases. The parameters of this formula can be
varied, taking into account current theoretical uncertainties in the reaction
rate. The results are illustrated for the example of the ^{12}C+^{12}C fusion
reaction. This reaction is very important for the understanding of nuclear
burning in evolved stars, in exploding white dwarfs producing type Ia
supernovae, and in accreting neutron stars. The S-factor at stellar energies
depends on a reliable fit and extrapolation of the experimental data. We
calculate the energy dependence of the S-factor using a recently developed
parameter-free model for the nuclear interaction, taking into account the
effects of the Pauli nonlocality. For illustration, we analyze the efficiency
of carbon burning in a wide range of densities and temperatures of stellar
matter with the emphasis on carbon ignition at densities rho > 10^9 g/cc.
