  Let $X=\C^n$. In this paper we present an algorithm that computes the de Rham
cohomology groups $H^i_{dR}(U,\C)$ where $U$ is the complement of an arbitrary
Zariski-closed set $Y$ in $X$. Our algorithm is a merger of the algorithm given
by T.~Oaku and N.~Takayama (\cite{O-T2}), who considered the case where $Y$ is
a hypersurface, and our methods from \cite{W-1} for the computation of local
cohomology. We further extend the algorithm to compute de Rham cohomology
groups with support $H^i_{dR,Z}(U,\C)$ where again $U$ is an arbitrary
Zariski-open subset of $X$ and $Z$ is an arbitrary Zariski-closed subset of
$U$. Our main tool is the generalization of the restriction process from
\cite{O-T1} to complexes of modules over the Weyl algebra. All presented
algorithms are based on Gr\"obner basis computations in the Weyl algebra.
