We provide a rigorous asymptotic analysis of adaptive Simpson quadratures for automatic integration of piecewise smooth functions. The standard quadrature collapses because the recursion does not terminate in the presence of non-trivial discontinuities. On the other hand, a properly selected subdivision strategy leads to an adaptive quadrature that almost surely asymptotically returns an approximation with the required accuracy. Moreover, it achieves the best asymptotic constant, which is the same as for integration of globally smooth functions.