  We consider cosmological dynamics in the theory of gravity with the scalar
field possessing the nonminimal kinetic coupling to curvature given as $\eta
G^{\mu\nu}\phi_{,\mu}\phi_{,\nu}$, where $\eta$ is an arbitrary coupling
parameter, and the scalar potential $V(\phi)$ which assumed to be as general as
possible. With an appropriate dimensionless parametrization we represent the
field equations as an autonomous dynamical system which contains ultimately
only one arbitrary function $\chi (x)= 8 \pi \vert \eta \vert V(x/\sqrt{8
\pi})$ with $x=\sqrt{8 \pi}\phi$. Then, assuming the rather general properties
of $\chi(x)$, we analyze stationary points and their stability, as well as all
possible asymptotical regimes of the dynamical system. It has been shown that
for a broad class of $\chi(x)$ there exist attractors representing three
accelerated regimes of the Universe evolution, including de Sitter expansion
(or late-time inflation), the Little Rip scenario, and the Big Rip scenario. As
the specific examples, we consider a power-law potential
$V(\phi)=M^4(\phi/\phi_0)^\sigma$, Higgs-like potential
$V(\phi)=\frac{\lambda}{4}(\phi^2-\phi_0^2)^2$, and exponential potential
$V(\phi)=M^4 e^{-\phi/\phi_0}$.
