An odd rational function with symbols in it is integrated in the square of the variable

[Rafael-SOWNet]

Part of #718.

A rational function with symbols in it, past the third degree below the bar, is now integrated in u = x² before the partial fractions run, when one side holds only odd powers of x and the other only even ones. x N(x²)/D(x²) dx becomes N(u)/(2 D(u)) du, and N(x²)/(x D(x²)) dx becomes N(u)/(2u D(u)) du. Factors written below the bar keep their shape. Where the odd side has an odd power of x among its written factors, one x is taken off that factor, so x^5 (d + g x²)(a + b x² + c x⁴) is read as x · u² (d + g u)(a + b u + c u²) rather than as a sextic in u that nothing factors.

Before this, partial fractions ran over the full polynomial in x, with symbolic coefficients. Twelve corpus rows timed out at 5 s on master 3c86dda7. They now answer:

integrand	Rubi	length	time
1/(x^5 (a + b x²)^10)	1.1.2.2	652	0.36 s
1/(x^7 (a + b x²)^10)	1.1.2.2	697	0.04 s
1/(x (a + b x² + c x⁴)³)	1.2.2.2	3,106	0.17 s
1/(x³ (a + b x² + c x⁴)³)	1.2.2.2	3,490	0.18 s
(A + B x²)/(x (a + b x² + c x⁴)³)	1.2.2.4	3,250	0.19 s
(A + B x²)/(x³ (a + b x² + c x⁴)³)	1.2.2.4	4,189	0.31 s
1/(x (d + g x²)(a + c x⁴)²)	1.2.2.4	2,870	0.09 s
1/(x^5 (d + g x²)(a + b x² + c x⁴))	1.2.2.4	3,045	0.05 s
(d + g x² + f x⁴)/(x³ (a + b x² + c x⁴)²)	1.2.2.6	2,657	0.16 s
(d + g x² + f x⁴)/(x^5 (a + b x² + c x⁴)²)	1.2.2.6	3,893	0.16 s
1/((d + g x)³ (a + b (d + g x)² + c (d + g x)⁴)³)	1.2.3.2	4,042	0.10 s
1/((d f + g f x)³ (a + b (d + g x)² + c (d + g x)⁴)³)	1.2.3.2	4,072	0.11 s

Lengths are characters of Stringize(). All twelve ran in one process, in this order.

Rubi corpus, both arms built from 3c86dda7, fix against master:

• Files 0–8, plus a 16,099-row pocket drawn mostly from families 1 and 4: the pocket goes from 15,858 to 15,870 solved, and its timeouts fall from 94 to 82. Family 1 gains one row; the other files have the same counts in both arms. Both arms have 0 wrong.
• Rows that moved: the twelve above. A rerun of only those rows gives 12/12 solved on this branch, graded by differentiating back, and 12 timeouts on master.
• Allocation gate: passed.
• Unit suite: 15,198 passed, 0 failed, 13 skipped (net10.0).

Where master already answered, the answers get shorter: (A + B x²)/(x³ (a + b x²)) goes from 656 characters to 174. In a 150-row sample of answers that changed, 15 got two to three times shorter and none changed verdict. BREAKING-CHANGES.md has an entry, An odd rational function with symbols in it is integrated in the square of the variable. AnOddRationalFunctionInTheSquareIntegralTest integrates three of the rows and checks each answer differentiates back.

This also takes the slowest named remainder in #718's renaming work, (k x + m x³ + n x⁵)/((d + g x²)(a + b x² + c x⁴)), from 19.7 s to 0.5 s. That branch waits for this one.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura