  We present a study of hydrogen at pressures higher than molecular
dissociation using the Coupled Electron-Ion Monte Carlo method. These
calculations use the accurate Reptation Quantum Monte Carlo method to estimate
the electronic energy and pressure while doing a Monte Carlo simulation of the
protons. In addition to presenting simulation results for the equation of state
over a large region of phase space, we report the free energy obtained by
thermodynamic integration. We find very good agreement with DFT calculations
for pressures beyond 600 GPa and densities above $\rho=1.4 g/cm^3$. Both
thermodynamic as well as structural properties are accurately reproduced by DFT
calculations. This agreement gives a strong support to the different
approximations employed in DFT, specifically the approximate
exchange-correlation potential and the use of pseudopotentials for the range of
densities considered. We find disagreement with chemical models, which suggests
a reinvestigation of planetary models, previously constructed using the
Saumon-Chabrier-Van Horn equations of state.
