Filon-Clenshaw-Curtis rules are among rapid and accurate quadrature rules for
computing highly oscillatory integrals. In the implementation of the
Filon-Clenshaw-Curtis rules in the case when the oscillator function is not
linear, its inverse should be evaluated at some points. In this paper, we solve
this problem by introducing an approach based on the interpolation, which leads
to a class of modifications of the original Filon-Clenshaw-Curtis rules. In the
absence of stationary points, two kinds of modified Filon-Clenshaw-Curtis rules
are introduced. For each kind, an error estimate is given theoretically, and
then illustrated by some numerical experiments. Also, some numerical
experiments are carried out for a comparison of the accuracy and the efficiency
of the two rules. In the presence of stationary points, the idea is applied to
the composite Filon-Clenshaw-Curtis rules on graded meshes. An error estimate
is given theoretically, and then illustrated by some numerical experiments