  Highly oscillatory integrals of composite type arise in electronic
engineering and their calculations is a challenging problem. In this paper, we
propose two Gaussian quadrature rules for computing such integrals. The first
one is constructed based on the classical theory of orthogonal polynomials and
its nodes and weights can be computed efficiently by using tools of numerical
linear algebra. We show that the rate of convergence of this rule depends
solely on the regularity of the non-oscillatory part of the integrand. The
second one is constructed with respect to a sign-changing function and the
classical theory of Gaussian quadrature can not be used anymore. We explore
theoretical properties of this Gaussian quadrature, including the trajectories
of the quadrature nodes and the convergence rate of these nodes to the
endpoints of the integration interval, and prove its asymptotic error estimate
under suitable hypotheses. Numerical experiments are presented to demonstrate
the performance of the proposed methods.
