A root of a power is not read as the power, and two proportional radicals are not a quotient of two

[Rafael-SOWNet]

Two wrong-answer classes found by the first sample of Rubi's family 4 (trigonometric functions), both on master, both fixed here; #1386 and #1387 are the issues. Cut from master; touches PolynomialSolver.ParseMonomial, the sin^p cos^q reader and the two-linear-radicals rule, and nothing the other open PRs touch.

#1387 — a root of a power is not the power. (sin^2)^(3/2) is |sin|^3, and two readers took it for sin^3:

• the polynomial parser read (x^q)^v as x^(q v) for any v, so through Simplify's long division (u^2)^(3/2)/u became u^2 and sqrt(u^2)/(-u) became -1 — wrong for every negative u, and a Simplify result, not only an integrator's. A whole v still multiplies ((x^q)^3 is x^q x^q x^q however x is signed); a fractional v over a first power is still read ((a x)^(3/2) as a^(3/2) x^(3/2), the generic reading every rule uses); over any other power it is refused — an odd one too, since (u^3)^(3/2) is i at u = -1 where u^(9/2) is -i. The parity-only guard was tried and rejected: it gave u^(7/2) for (u^3)^(3/2)/u, the same class of wrong answer one sign along.
• the sin^p cos^q reader multiplied the same way; a fractional power of anything holding an even whole power of a trigonometric function is refused.

Six family-4 answers were wrong by it, on every other half-turn. Two are now answered with the sign in them ((a sin^2)^(5/2), 1/sqrt(a cot^2) — sgn(sin x) …, by the sign-of-the-complement route) and four are declined: the modulus of a trigonometric function is a class with no rule yet, and unevaluated beats wrong.

#1386 — sin(a + b (c + d x)^(1/3))/(c e + d e x)^(1/3) integrated to 0 provided …. The rule that rationalises two fractional powers of different linears by the quotient of their roots took two proportional linears; their determinant is zero, the second linear in t was 0/(a − c t^q), and everything it multiplied vanished before anything was asked in t. A zero determinant, exactly or once simplified, declines.

Recorded in BREAKING-CHANGES.md, with a seventh wrong answer on the independent suites — asin(sqrt(1 − x^2))/sqrt(1 − x^2) was answered −asin(…)^2/2, right for x > 0 and off by a sign for x < 0, the same identity through sqrt(x^2) — and the three answers the strict reading costs, each right where the integrand is real or for a positive symbol, and only there: (b tan^3)^(3/2), integrated as b^(3/2) tan^(9/2) (the odd-power reading; (u^3)^(3/2) is i at u = -1 where u^(9/2) is -i, so the two agree only where the tangent is positive, which is where the integrand is real); asin(x/a)^(3/2)/sqrt(a^2 − x^2), whose derivative-divides step collected a sqrt(1 − x^2/a^2) to sqrt(a^2 − x^2) through (a^2)^(1/2) = a; and sec(c+dx)^(3/2)/(b sec(c+dx))^(5/2), which Simplify took to cos/(b^(5/2) d) through (1/cos)^(3/2) = cos^(-3/2), wrong for every negative cosine and cancelling against the same wrong step on the denominator. The reader that could restore the last two soundly is a numeric proportionality check in the substitution search rather than a symbolic collection, which is a follow-up.

The independent suites: 1707 → 1705 (the two asin rows), 0 wrong, 0 timeouts. Family 4's sample (422 problems at 6 per file): 7 wrong → 0 wrong, 220 → 220 solved (the two answered with the sign against the three recorded), 48 → 49 timeouts. Four suites green; the new tests check each of the six and the tangent at points on both sides of a zero of the function, and the Simplify rows at negative u.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura