  In this paper we revisit some quadrature methods for highly oscillatory
integrals of the form $\int_{-1}^1f(x)e^{{\rm i}\omega x}dx, \omega>0$.
Exponentially Fitted (EF) rules depend on frequency dependent nodes which start
off at the Gauss-Legendre nodes when the frequency is zero and end up at the
endpoints of the integral when the frequency tends to infinity. This makes the
rules well suited for small as well as for large frequencies. However, the
computation of the EF nodes is expensive due to iteration and ill-conditioning.
This issue can be resolved by making the connection with Filon-type rules. By
introducing some $S$-shaped functions, we show how Gauss-type rules with
frequency dependent nodes can be constructed, which have an optimal asymptotic
rate of decay of the error with increasing frequency and which are effective
also for small or moderate frequencies. These frequency-dependent nodes can
also be included into Filon-Clenshaw-Curtis rules to form a class of methods
which is particularly well suited to be implemented in an automatic software
package.
