  Adding linear combinations $R^2,R_{\mu\nu}R^{\mu\nu}$ and
$R_{\mu\nu\eta\delta}R^{\mu\nu\eta\delta}$ with Einstein-Hilbert action we
obtain interior metric of an an-isotropic spherically symmetric collapsing
(ASSC) stellar cloud. We assume stress tensor of the higher order geometrical
terms to be treat as an-isotropic imperfect fluid with time dependent density
function $\rho(t)$ and radial and tangential pressures $p_r(t)$ and $p_t(t)$
respectively. We solved linearized metric equation via perturbation method and
obtained 12 different kinds of metric solutions. Calculated Ricci and
Kretschmann scalars of our metric solutions are non-singular at beginning of
the collapse for 2 kinds of them only. Event and apparent horizons are formed
at finite times for two kinds of singular metric solutions while 3 metric
solutions exhibit with event horizon only with no formed apparent horizon.
There are obtained 3 other kinds of the metric solutions which exhibit with
apparent horizon with no formed event horizon. Furthermore 3 kinds of our
metric solutions do not exhibit with horizons. Barotropic index of all 12 kinds
of metric solutions are calculated also. They satisfy different regimes such as
domain walls (6 kinds), cosmic string (2 kinds), dark matter (2 kinds),
anti-matter (namely negative energy density) (1 kind) and stiff matter (1
kind). Time dependent radial null geodesics expansion parameter $\Theta(t)$ is
also calculated for all 12 kinds of our metric solutions. In summary 4 kinds of
our solutions take absolutely positive value $\Theta>0$ which means the
collapse ended to a naked singularity but 8 kinds of our metric solutions ended
to a covered singularity at end of the collapse where $\Theta\leq0$ and so
trapped surfaces are appeared.
