A symbolic exponent that is -1 as a value is integrated to the logarithm

[Rafael-SOWNet]

Part of #718.

x^(-1 - 3n) (a + b x^n)^3 came back as a sum whose last term was x^0/0, so that the answer had no value anywhere, and x^(-n/n) as NaN + C. Both were so on 2.5.0 too:

integrand	master 51cb049a	this
x^(-1 - 3n) (a + b x^n)^3	three powers and x^0/0, no value anywhere	the three powers and b^3 ln(x)
x^(-1 - 3n + 3n)	x^(-1 - 3n + 3n + 1)/(-1 - 3n + 3n + 1), no value anywhere	ln(x)
x^(-n/n)	NaN	ln(x)
1/(a (b x^m)^n)^(1/(m n))	NaN	(a b^n)^(-1/(m n)) ln(x), provided b > 0 and a b^n > 0

What changes. Expanded, x^(-1 - 3n) (a + b x^n)^3 leaves x^(-1 - 3n + 3n) for its last term, which is x^(-1) for every n. The power rule reads -1 once the exponent is the number (#1258's fix), but InnerSimplified keeps -3n + 3n as two terms, and the rule read this one as written and divided by its zero. It asks whether the exponent plus one vanishes as a value now -- at pinned values first and then simplified, as the integrator asks elsewhere -- and gives the logarithm where it does. A symbolic exponent that is -1 for one value only, x^n itself, is the power rule's as before.

Tests: ExpandedSquareNaNTest.AnExponentThatIsMinusOneAsAValue, five rows, each differentiated back with the symbols pinned; all five fail on master. x^n and x^(n - 1) join ThePowerRuleIsUnchanged.

Found by work/intbench, which counts an answer that has no value anywhere as wrong, on every fourth problem of Rubi's 1.1.2.2 and 1.1.3.2: five there, the four rows of this shape and the last row above. Its usual sample of forty a file reaches none of them.

Measured first on every sixth of the 6,519 problems in the corpus with a symbolic exponent, 1,087, at the corpus's 5-second budget, against master 8f3757cd, the branch's base:

	master	this
solved	944	947
wrong	3	0

The three are 1.1.3.2's 1/(a x^n)^(1/n), x^(-1 - 4n) (a + b x^n)^5 and x^(-1 - 8n) (a + b x^n)^8.

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1297
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the sample on either build. Of the eight problems the two builds disagreed on, run again one build at a time, three are the pocket's wrong answers, right here, and the other five are answered on both.

The suite passes on the commit measured, a0c1e98e, 14,598 tests, and so does the allocation gate. Master 51cb049a is merged in since, without conflicts; the calculus tests pass on the merge, 3,990 of them, and the four rows above are right on it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura