Integrate a rational function over an inverse tangent's quadratic beside it by partial fractions

[Rafael-SOWNet]

Part of #718.

R(x) atan(a x) over c + a^2 c x^2: the rational factor is divided down first, with the provided condition stripped from the remainder. It is then split over the quadratic and the powers of x by partial fractions, and each part beside the inverse tangent is integrated on its own. Before, x^4 atan(a x)/(c + a^2 c x^2) and three like it ran out of time or were declined.

Rubi chain (master arm at 67d8e8d, 5 s budget):

• Pocket (5.3 and 5.4, 800 rows): master 749, this branch 753.
• Families 0–8 sampled: identical on both arms.
• Moved rows run alone: master solves 0 of 4, the branch 4 of 4. They are 5.3.4:216, 218, 222 and 224.
• Gate: passed.
• Suite: the chain's suite run was killed by the local memory guard. That was the spike fixed in Collect the documented examples' compilations as the test runs them #1920, which this branch did not yet have. After merging master, the full suite passes: 15,527 passed.

On 2.5.0 all four were integral(...). The branch answers them in 126 to 894 characters, each in about 0.4 s.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura