  We consider a scalar-tensor model of dark energy with Gauss-Bonnet and
non-minimal couplings. Exact cosmological solutions were found in absence of
potential, that give equations of state of dark energy consistent with current
observational constraints, but with different asymptotic behaviors depending on
the couplings of the model. A detailed reconstruction procedure is given for
the scalar potential and the Gauss-Bonnet coupling for any given cosmological
scenario. Particularly, we consider conditions for the existence of a variety
of cosmological solutions with accelerated expansion, including quintessence,
phantom, de Sitter, Little Rip. For the case of quintessence and phantom we
have found a scalar potential of the Albrecht-Skordis type, where the potential
is an exponential with a polynomial factor.
