  We give an algorithm to compute the following cohomology groups on $U = \C^n
\setminus V(f)$ for any non-zero polynomial $f \in \Q[x_1, ..., x_n]$; 1.
$H^k(U, \C_U)$, $\C_U$ is the constant sheaf on $U$ with stalk $\C$. 2. $H^k(U,
\Vsc)$, $\Vsc$ is a locally constant sheaf of rank 1 on $U$. We also give
partial results on computation of cohomology groups on $U$ for a locally
constant sheaf of general rank and on computation of $H^k(\C^n \setminus Z,
\C)$ where $Z$ is a general algebraic set. Our algorithm is based on
computations of Gr\"obner bases in the ring of differential operators with
polynomial coefficients.
