Due to the increasing number of objects in Low Earth orbit, the fast and reliable estimation of the collision risk is an important challenge for spacecraft owners/operators. Among the available risk indicators, we focus on computing the instantaneous probability of collision, which can be modeled as the integral of a three-dimensional Gaussian probability density function over a Euclidean ball. We propose an efficient and accurate method for evaluating this integral. It is based on the combination of two complementary strategies. For the first one, convergent series and numerical error bounds are computed. These bounds provide a tradeoff between the accuracy needed and the number of terms to compute. The second one, using divergent series, approximates the value of the integral with a good accuracy in most cases with only a few terms computed. Based on those two methods, a hybrid algorithm is tested on cases borrowed from the literature and compared against existing methods. Several numerical results and comparisons confirm both the efficiency and robustness of our approach.