Singular and nearly singular integrals involving Cauchy and log kernels of weighted orthogonal polynomials solve simple three-term recurrences. By a careful choice between forward recurrence and (F.W.J.) Olver's algorithm we can efficiently and accurately compute these singular integrals, arbitrarily close or even on the element. These techniques extend to log kernel integrals on the square where one needs to simultaneously utilise two bivariate recurrence relationships.