Numerical quadrature methods for irregular oscillatory integrals for the form \vint_a^b f(x) g (\omega, x)\, \hboxdx are now being developed for oscillatory functions g ( , x ) which have the form e i q(x) and J n ( q ( x )) where the function q ( x ) is the irregular argument and the oscillatory frequency. It is demonstrated here that such rules can be found from simple integration by parts with some innovative manipulation in the Bessel function case.

The generated rules are illustrated with numerical experiments, and yield excellent practical convergence.