  We develop two classes of composite moment-free numerical quadratures for
computing highly oscillatory integrals having integrable singularities and
stationary points. The first class of the quadrature rules has a polynomial
order of convergence and the second class has an exponential order of
convergence. We first modify the moment-free Filon-type method for the
oscillatory integrals without a singularity or a stationary point to accelerate
their convergence. We then consider the oscillatory integrals without a
singularity or a stationary point and then those with singularities and
stationary points. The composite quadrature rules are developed based on
partitioning the integration domain according to the wave number and the
singularity of the integrand. The integral defined on a subinterval has either
a weak singularity without rapid oscillation or oscillation without a
singularity. The classical quadrature rules for weakly singular integrals using
graded points are employed for the singular integral without rapid oscillation
and the modified moment-free Filon-type method is used for the oscillatory
integrals without a singularity. Unlike the existing methods, the proposed
methods do not have to compute the inverse of the oscillator. Numerical
experiments are presented to demonstrate the approximation accuracy and the
computational efficiency of the proposed methods. Numerical results show that
the proposed methods outperform methods published most recently.
