An algorithm is described for the efficient and reliable evaluation of badly behaved definite integrals to a prescribed accuracy by concentrating the abscissas near the regions of greatest irregularity in the integrand. This is achieved by subdividing the interval of integration and by using a combination of the 7-point Clenshaw–Curtis quadrature and the 9-point Romberg quadrature in each subinterval. We argue that our algorithm will nearly minimise the number of function evaluations needed to evaluate a badly behaved integral.