Here is a revised version with shorter sentences, no “we,” and more natural English phrasing: **Revised Summary** This work addresses symbolic integration in finite terms, with a focus on limitations of the Risch–Norman algorithm. Traditional heuristic degree bounds often fail for integrands that involve transcendental non-elementary functions. To overcome this problem, we formalize complete reduction systems and present a refined algorithm based on conditional identities and reduction rules. Key concepts include precompleteness and completeness of reduction systems relative to a semigroup order. The refined process is defined to resolve critical pairs by splitting rules and reducing intermediate identities. This refinement avoids the non‑termination issues found in Norman's original approach in some cases. It also successfully handles examples where standard methods fail. The work provides a rigorous foundation for a more reliable reduction‑based symbolic integration algorithm.