Generalizing the Cramér-von Mises and the Kolmogorov-Smirnov test, different integral statistics based on L_p-norms are compared with respect to local approximate Bahadur efficiency. Simulation results corroborate the theoretical findings. Several examples illustrate that goodness-of-fit testing based on L_p-norms should receive more attention. It is shown that, given a distribution function F₀ and a specific alternative, one can draw the plot of efficiency as a function of p and determine the value of p giving the maximum efficiency.