This article concerns a comparison of several tests for testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous Poisson process (NHPP). The study includes the well-known Laplace test statistic, the most powerful test for the shape parameter in a Poisson process with Weibull intensity, the likelihood ratio test against arbitrary NHPP alternatives, two nonparametric tests for trends based on Kendall's tau and Spearman's rho, and a test based on an F statistic. The powers of the tests are determined by Monte Carlo simulation against alternatives that are increasing at an exponential rate, a power rate (Weibull intensity), and a logarithmic rate. Alternatives that are step functions with one jump are also considered. In a few cases, the exact powers are also obtained analytically.