In symbolic integration, the Risch--Norman algorithm aims to find closed forms of elementary integrals over differential fields by an ansatz for the integral, which usually is based on heuristic degree bounds. Norman presented an approach that avoids degree bounds and only relies on the completion of reduction systems. We give a formalization of his approach and we develop a refined completion process, which terminates in more instances. In some situations when the algorithm does not terminate, one can detect patterns allowing to still describe infinite reduction systems that are complete. We present such infinite systems for the fields generated by Airy functions and complete elliptic integrals, respectively. Moreover, we show how complete reduction systems can be used to find rigorous degree bounds. In particular, we give a general formula for weighted degree bounds and we apply it to find tight bounds for above examples.

翻译：在符号积分中，Risch--Norman 算法旨在通过待定积分法（通常基于启发式次数上界）寻找微分域上初等积分的闭形式。Norman 提出了一种避免次数上界、仅依赖约化系统完备化的方法。我们对其方法进行了形式化描述，并发展了一种更精细的完备化过程，该过程在更多情况下可终止。当算法无法终止时，可通过检测模式来描述仍然完备的无限约化系统。我们分别针对由 Airy 函数和完全椭圆积分生成的域给出了此类无限系统。此外，我们展示了完备约化系统如何用于建立严格的次数上界。特别地，我们给出了加权次数上界的一般公式，并将其应用于上述示例以求得紧密上界。