AI 中文总结
本文评估SymPy中并行Risch-Norman方法在Charlwood 50个积分上的表现，49个成功，1个证明非初等，并分析性能瓶颈与优化方向。
AI 中文摘要
我们使用Charlwood 2008年套件中的50个具有挑战性的不定积分，评估了并行Risch-Norman方法的SymPy实现。该系统自动构建被积函数塔，通过微分验证答案，并为49个问题返回正确、已验证的积分，零错误。唯一的失败案例$\u222b\rcsin(x\u221a(1-x^2))\,dx$被证明是非初等的，使用了全纯余项证书，尽管受限于亏格三曲线上的未验证完整性假设。第二部分的次数界成功地将经典待定形式大小减少了三分之二，且不影响运行时间。论文详细介绍了算法机制，如$S'$-单位、Pell单位和塔坐标中的留数计算。与成熟实现相比，这个未经调优的SymPy原型比FriCAS慢（中位积分慢六倍），但比AXIOM更准确（AXIOM返回了三个错误答案）。性能剖析指出了非线性范数搜索和嵌套数域中的性能瓶颈，概述了优化的明确目标。
英文摘要
We evaluate a SymPy implementation of the parallel Risch-Norman method using Charlwood's 2008 suite of 50 challenging indefinite integrals. The system automatically builds integrand towers, verifies answers by differentiation, and returns correct, verified integrals for 49 problems with zero errors. The single failure, $\int\arcsin(x\sqrt{1-x^2})\,dx$, is proven non-elementary using a holomorphic-remainder certificate, though limited by an unverified completeness hypothesis on a genus-three curve. Part II's degree bounds successfully reduce classical ansatz sizes by two-thirds without impacting running time. The paper details algorithmic mechanisms like $S'$-units, Pell units, and residue computing in tower coordinates. Compared to mature implementations, this untuned SymPy prototype is slower than FriCAS (by a factor of six on the median integral) but more accurate than AXIOM (which returned three wrong answers). Profiling pinpoints performance bottlenecks in nonlinear norm searches and nested number fields, outlining clear targets for optimisation.
