Benchmarking the USM Against Mathematica Integrate, Part 3

This report compares the Unified Substitution Method (USM) with Mathematica’s native `Integrate` function across five families of integrals involving quadratic radicals, circular and hyperbolic substitutions, and trigonometric half-angle structures. For each transform family, the report includes the Wolfram benchmark code, timing data, expression-size measurements, derivative-based verification tests, and commentary on timeouts, unevaluated outputs, branch behavior, and expression swell.

The strongest result appears in Transform 1, where, on the six cases solved by both methods, Mathematica was about 194× slower in cold timing, 380× slower in warm timing, and produced closed-form outputs roughly 753× larger by byte count. In aggregate, USM produced 95 non-timeout antiderivatives out of 100 examples, with 94 fully verified, while Mathematica produced closed outputs for only 6 of the 10 paired examples; those native closed outputs averaged about 4.45 MB, compared with roughly 4.6 KB for USM. 

The report also highlights several complementary performance patterns. Transform 2 demonstrates coverage and predictability: USM solved and verified 100/100 examples, while Mathematica closed only 4/10 paired examples, with additional timeouts and unevaluated integrals.  Transform 4 gives another strong expression-swell result: USM solved and verified 20/20 examples, while Mathematica closed 6/10, and on the mutually closed subset Mathematica was about 110× slower cold, 129× slower warm, and produced outputs more than 100× larger on average. 

Overall, the benchmarks show that the USM framework is especially effective when direct Mathematica integration leads to timeouts, unevaluated integrals, or very large antiderivative expressions. At the same time, the report documents cases where Mathematica remains competitive, particularly in the simpler hyperbolic radical family of Transform 3 and in some successfully closed Transform 5 examples. This makes the comparison useful as a technical benchmark reference rather than a one-sided performance claim. Across the five families, USM solved and verified all examples in Transforms 2–5, and nearly all of Transform 1, supporting the claim that USM converts the targeted radical and half-angle families into stable rational-in-parameter integrations with improved coverage, predictability, and, in several cases, substantial suppression of expression swell.