The restricted three-body problem (RTBP) has in the past played an essential role in many different areas of dynamical astronomy, and indications are that this will continue. As the state of the art in computing becomes more advanced, larger numbers of integrations and longer durations are attempted. Thus, computational efficiency and accuracy are becoming more important. Also, the use of the RTBP in many different areas leads to the desire for a general integration method. In order to maximize the efficiency of orbit calculations, comparisons are made of different methods of integration. The results can be summarized as follows: 1. The Bulirsch-Stoer extrapolation method is extremely fast and accurate, and is the method of choice. 2. Regularization of the equations of motion is essential. 3. When applicable, a manifold correction algorithm, originally due to Nacozy (1971), reduces numerical errors to the limits of machine accuracy, and at a cost of only 1 to 3 percent in cpu time.