The computation of indefinite integrals in certain kind of “closed form”, which is known as symbolic integration, is an important research subarea of computer algebra. After implementing the recursive Risch algorithm partly, it was realized that efficient algorithms can be achieved by a parallel approach. This led to the famous Risch–Norman algorithm. However, this approach relies on an ansatz with heuristic degree bounds. Norman’s completion-based approach provides an alternative for finding the numerator of the integral avoiding heuristic degree bounds. However, depending on the differential field and on the selected ordering of terms, it may happen that the completion process does not terminate and yields an infinite number of reduction rules. We apply Norman’s approach to the differential fields generated by Airy functions, which play an important role in physics. By fixing adapted orderings and analyzing the process in the concrete case, we present two complete reduction systems for Airy functions by finitely many formulae to denote infinitely many reduction rules.