[Rafael-SOWNet]

Part of #233.

On Rubi's 8.5 (the hyperbolic integral functions) and 4.3.3.1, four routes spent gigabytes declining before the one that answers.

1. SolveBySplittingSum expanded Shi(u) or Chi(u) times (e^u ± e^-u)/x^n into exponential terms, each non-elementary on its own. It declined after 10–13 GB, and SolveIntegratingByParts then answered the whole in 0.2–0.9 GB. IsAHyperbolicIntegralBesideItsExponentialsOverAPolynomial now sends such an integrand to by-parts first, the same call at the same depth, as the rule for an inverse function times a rational one already does.
2. SolveByScalingTheVariable ran three full Simplify calls before finding that x = c t leaves c inside e^(b x) or Shi(u): about 4.6 of the 5.2 GB of x Shi(b x)^2. TheScaleStaysInsideAFunction evaluates the scaled integrand at two scales and two points first. It can only decline; where the values agree, the symbolic check runs as before.
3. SolveBySubstitution simplified each quotient by a special function's derivative, about a gigabyte each, three times for x^2 Chi(a + b x)^2. A special function is transcendental over the elementary functions, so if the quotient takes two values at two points with the function held, no rewriting removes x from it. VariesWhereASpecialFunctionIsHeld refuses it before the simplification. It complements Differentiate what does not mention the variable to 0, whatever its node #2014's KeepsTheVariableThroughEveryRewriting, which leaves special functions out.
4. SolveByTangentSubstitution's cot^k leaves 1^k above the bar and u^k below. That fell outside Skip simplifications that cannot help in the tangent rule and the substitution search #2002's lowest-terms class, which skips Simplify, and a full Simplify spent 4.8 GB dropping the 1^7. It is in that class now.

The first mechanism this branch had, stopping a split at its first declined term, merged as #2016 meanwhile; the rebase took master's side.

Answers are identical.

Chain against master f7f2a7a, which already has #2015's Simplify change, with the harness at its current revision on both arms:

• pocket of 3,919 rows (4.5.x, 4.3.3.x and 8.x every 2nd row, the rest of 4.3 and 4.4 every 6th, every 30th elsewhere): verdicts identical;
• allocation over the rows solved on both: 460.9 → 435.0 GB; 8.5 alone: 31.0 → 11.8 GB;
• samples of families 0–8: unchanged in verdicts; family 8's allocation 7.9 → 6.4 GB;
• no row's allocation grew past 1.3×.

Per row, by the probe against d3a6bce, before the rebase:

row	before	this
8.5:62 cosh Shi/x^2	12.59 GB	0.17 GB
8.5:154 Chi sinh/x^2	12.68 GB	0.19 GB
8.5:21 x Shi(b x)^2	5.21 GB	0.46 GB
4.3.3.1:313	5.01 GB	0.20 GB

A new class pins each shape: 8 cases differentiate back, and 7 more are bounded at 2 GB on the test's own thread, which they exceed (3.5–23.5 GB) without these changes. Locally, 1,166 tests pass across that class, the hyperbolic, substitution and tangent classes, and the corpus gate.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura