  In this work we propose and analyse a numerical method for computing a family
of highly oscillatory integrals with logarithmic singularities. For these
quadrature rules we derive error estimates in terms of $N$, the number of
nodes, $k$ the rate of oscillations and a Sobolev-like regularity of the
function. We prove that that the method is not only robust but the error even
decreases, for fixed $N$, as $k$ increases. Practical issues about the
implementation of the rule are also covered in this paper by: (a) writing down
ready-to-implement algorithms; (b) analysing the numerical stability of the
computations and (c) estimating the overall computational cost. We finish by
showing some numerical experiments which illustrate the theoretical results
presented in this paper.
