Computer algebra: the textbook-steps run

Computer algebra systems cover far more mathematics than this comparison: their scope is not in question here. This page is about applying textbook rules step by step and about what certifies the answer. For the broader question of what tools change for a model solving science tasks, see Science tasks: models with and without tools.

Result: a bank of 129 checked rows — derivative, integral, limit and ODE rows (dr-1-*, di-1-*, li-1-*, od-1-*) — was recounted on 2 October 2026. SymPy 1.14.0 and Wolfram 15.0.1 (for Linux x86, 64-bit) agree on all 129; Arxo, checking the presented steps, records zero false accepts and zero silent refusals on expected cases. On top of the bank, 15 textbook cases C01–C15 from OpenStax Calculus (editions pinned in the package sources) were mapped against both references: C01–C05 were executed by the Arxo case package (6 of 6 tests pass, including the related by-parts case C13r); the remaining cases sit on the stated model boundary, except C14, where the two references disagree. The numbers belong to SymPy 1.14.0, Wolfram build 15.0.1, and law 0.1.1.

The task and the reference

The expectations come from a printed textbook, OpenStax Calculus, not from either implementation: each case states the rule to apply (power rule, sine derivative, power antiderivative, net change, a special limit, series, corner cases) and the value the textbook requires. SymPy 1.14.0 serves as the main reference; Wolfram 15.0.1 serves as the second reference, called fresh 15 times for the C01–C15 mapping (one probe plus 14 case calls, no cache reuse). The Arxo side applies the pinned textbook formulations through the packages openstax.calculus_derivatives, openstax.calculus_integrals, openstax.calculus_limits and openstax.calculus_series. Public pages of the references are at SymPy and the Wolfram Language.

What “agreement” means here

Three outcome classes are used. A case matches when both sides give the textbook value, possibly in different but equivalent forms (factored or expanded, an unevaluated-input refusal against an explicit infinity where both mean divergence). A case is not comparable when the two sides answer different questions by construction (a rule the catalog does not state). C14 alone is a mismatch between a reference and the textbook, described below. An honest refusal — no value with a stated reason — is its own class, not a failure.

Results

Bank: 129 of 129 rows agree between SymPy 1.14.0 and Wolfram 15.0.1; Arxo false accepts 0 of 129. Cases C01–C05 below are a subset of those rows and were additionally executed by the Arxo case package.

Case	Expectation (OpenStax)	SymPy 1.14.0	Wolfram 15.0.1	Arxo	Outcome
C01	3x squared	3x squared (bank dr-1-0000)	confirmed 1/1 in bank	PASS, power rule justifies (executed)	match
C02	cos x	cos x (bank dr-1-0018)	confirmed 1/1 in bank	PASS, sine rule justifies (executed)	match
C03	x cubed over 3 plus constant	x cubed over 3 (bank di-1-0000)	confirmed 1/1 in bank	PASS, power antiderivative justifies (executed)	match
C04	one half	one half	one half	PASS, net change justifies the definite integral (executed, exact case)	match
C05	1	1 (bank li-1-0024)	1 in bank	PASS, special-limit rule justifies (executed; mapping note in force)	match, with note
C06	diverges	infinity	input returned unevaluated	outside the model (no case of this shape)	match in substance, forms differ
C07	1/a for positive real part of a; otherwise not established	piecewise: 1/a under condition, else the integral form	conditional form with Re(a) > 0; with a > 0 gives 1/a	outside the model (the catalog states no conditional branches)	match (the two conditions are equivalent)
C08	pi squared over 6	pi squared over 6	Pi squared over 6	outside the model (no series family in the case bank)	match
C09	diverges	infinity	sum returned unevaluated	outside the model (as C06)	match in substance, forms differ
C10	does not exist (corner point)	no value (not-a-number at zero; a complex form, not a value)	derivative left unevaluated	outside the model (no corner-point rule)	match (neither side gives a value)
C11	(x-1)(x+1)	factored form; expansion returns the input	the same product, factors ordered differently	polynomial-factor shoulder, not executed here	match
C12	2 pi i k	integer-indexed set of values	integer-parameter form	outside the model (no solver package)	match (forms equivalent)
C13	e to x times (sin x minus cos x) over 2	equal value, expanded	equal value	exact case not executed; related by-parts case C13r PASS	match (values equal; the decomposition choice belongs to the student)
C14	1/(1-r) for |r| < 1; otherwise not established	piecewise with the |r| < 1 condition	the bare formula without the condition; at r = 1/2 gives 2	outside the model (as C08)	mismatch between Wolfram and the textbook
C15	diverges	infinity	input returned unevaluated	outside the model (as C06)	match in substance, forms differ

Where the two systems behave differently

One case, C14, the geometric series: SymPy 1.14.0 returns the sum guarded by the condition |r| < 1, which is what the textbook requires; Wolfram 15.0.1 returns the bare closed form without the condition. At r = 1/2 both references give 2, so the difference only matters where the condition decides the answer. In a bank pipeline such a case would leave the expected set and enter reference disagreement — 1 of 9 mapped cases here, with the other 8 agreeing in substance, including honest refusals in different forms (C06, C09, C10, C15).

A second, quieter difference is form, not value: where the textbook expects divergence, SymPy writes infinity while Wolfram returns the input unevaluated (C06, C09, C15); where no value exists, SymPy surfaces not-a-number or a complex form while Wolfram leaves the derivative unevaluated (C10). The contract counts these as agreement in substance.

Where the model stops

Series (C08, C14), improper integrals (C06, C09, C15), parametric cases with assumption branches (C07), the corner point (C10) and complex roots (C12) sit outside the checked catalog: where the catalog states no rule, Arxo answers “not established” with the reason instead of a value. That silence with a reason is a property of the package, stated in advance — the same property the bank measures on its negative controls. One further limit belongs to all three sides equally: no side ships an independent certificate with its answer. The bank answers are verdict records, and the cross-check is Wolfram times SymPy independently, 129 of 129 — not a proof object either side could hand over.

Reproducing the run

Pin the bank at its recorded commit and recount the five per-family answer files (wolfram, sympy, expected lines) against each other; that comparison is the only execution this run adds, and it uses the plain interpreter without the bank tooling. Pin the Arxo side: law 0.1.1 with the OpenStax packages named above; the C01–C05 and C13r cases execute as package tests (6 of 6 pass). Pin the mapping calls: SymPy 1.14.0 locally, Wolfram build 15.0.1, 15 fresh calls in total. Each mapped answer is filed verbatim with its case identifier and reference version, so any row of the table can be rechecked case by case.