Take a whole power out of a power of a product, in front of a constant

[Rafael-SOWNet]

(c R f^n)^r is K f^(n r) R^r with K = (c R f^n)^r/(f^(n r) R^r). The logarithmic derivatives of both sides are equal, so K is constant wherever it is defined. It stands in front of the answer the way Rubi writes (c sin(a + b x)^3)^(1/3) csc(a + b x). Only a factor of the integrand is rewritten, never a root inside another function, and only at the question asked.

The constant-splitting rule also stops adding provided c > 0 for a root with an odd denominator. The library takes the real root there, so (c sin(a + b x)^3)^(1/3) is real for c < 0 as well (52 → 35 chars on master f68bebd).

Corpus, measured against master f68bebd:

Set	master	this
pocket (1.3.2, 4.1.12, 1.2.2.3, 4.1.0, 4.1.7, 4.1.1.2)	2,045 / 2,160	2,060 / 2,160
families 0–8, sampled	—	equal, except 4.1 +1 and 6.4 +1

• 16 moves, all Unsolved → Solved: 1.3.2:1192, 1193 and 1248; 4.1.12:426–428, 434, 436, 446, 458–460, 466, 468 and 478; and 6.4.7:37.
• 0 wrong, and no move the other way. The gate and suite pass.
• The new answers are 35–391 chars.
• Each answer was checked exactly at points on both signs of sin and of x, with c = 2, −3 and 1/5. The residual is 0 at 50 and 2,000 digits.

Part of #718

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura