[RuleBased] change of variables loses the domain of the substituted variable, breaking ∫sin(sqrt(1+x))/sqrt(1+x)dx #145

∫ sin(sqrt(1+x))/sqrt(1+x) dx no longer closes with the rule-based method. It is entry 15
of the difficult baseline, and it fails in every difficult CI job of PRs #136, #138 and
#141 — on all three independently, and also on clean main with a current dependency
resolution, so it is not caused by any of those PRs:

[ fail ]∫( sin(sqrt(1 + x)) / sqrt(1 + x) )dx = -2cos(sqrt(1 + x)) but got: ...

Mechanism

Rule 4_1_12_81 (src/methods/rule_based/rules2/4 Trig functions/4.1 Sine/4.1.12 (e x)^m (a+b sin(c+d x^n))^p.jl:562)
changes variables to u = (e + f*x)^n via int_and_subst, which integrates in u and then
substitutes back with a plain substitute(result, from => to)
(src/methods/rule_based/rules_utility_functions.jl:466-485).

Nothing records that u is a principal square root when n = 1//2, and therefore
nonnegative. The integrand rebuilt in u contains sqrt(u^2):

• Until SymbolicUtils 4.46.7, simplify folded sqrt(u^2) to u, the rule closed, and the
  test passed — but that fold was unsound for negative u, which was
  (x^2)^(1//2) folds to x, dropping the absolute value for negative real x SymbolicUtils.jl#1045.
• Since 4.46.8, the POW_RULES entry @rule(sqrt((~x)^2) => abs(~x)) (fix: restrict nested power folding (x^a)^b to integer b to prevent unsound simplifications SymbolicUtils.jl#1077)
  correctly produces abs(u) instead. No rule cancels that abs(u) against the u in the
  numerator, so contains_int(result) stays true and the antiderivative is returned as an
  unevaluated substitute_after_int placeholder.

The upstream rule is correct. The gap is on this side: the change of variables discards the
domain of the substituted variable. JuliaSymbolics/SymbolicUtils.jl#1046 (no simplification
rules for abs) is why nothing removes the abs afterwards.

Possible fixes

1. Carry the domain of the substituted variable through int_and_subst, so that u >= 0 is
   available to simplification. The VariableDomain metadata key proposed in
   feat: add VariableDomain metadata to support mathematical assumptions Symbolics.jl#1983 would give a place to record it.
2. More narrowly, have int_and_subst rewrite sqrt(u^2) to u when it knows the
   substitution it introduced is nonnegative.

Either way a regression test on this integrand would be welcome, since the baseline entry
alone did not make the cause obvious.

Provenance of the evidence

The CI failures and the rule/int_and_subst source paths above were read directly. The
4.46.6 → 4.46.7 → 4.46.8 boundary was established by the automated reviews of #138, #140 and
#141 rather than re-bisected here; the commit e03f3da4 that adds the abs rule is in
SymbolicUtils PR #1077, which is consistent with that boundary.

Assisted by: AI