A power of the cosine beside a secant binomial is integrated in the cosine

[Rafael-SOWNet]

A power of the cosine beside a secant binomial, cos(c + d x)^2/(a + b sec(c + d x)), is one quotient in the cosine, cos(c + d x)^3/(a cos(c + d x) + b), wherever the secant is defined. The rule that rewrites a rational function of the secant in the cosine now takes it when the cosine of the same argument stands beside the secant. The same holds for the sine beside a cosecant. Part of #718.

The rule fires only when a sum in the secant is below the bar, and only to the first power:

• To the first power only. cos(x)/(a + b sec(x))^4 in the cosine is a partial-fraction search past five seconds, while the secant's own rules answer it in a third of one.
• A sum in the secant only. csc(x)^3/(a + b sin(x)^3) belongs to the rules for the sine's binomial (A sine's binomial cube below the bar is split over the cube's two factors #1850).

Measured on master 5dad421 and this branch:

Integrand	2.5.0	master	this branch
cos(c+d*x)^2/(a+b*sec(c+d*x))	declined	117,643 chars	1,091 chars
cos^2 (A+B sec)/(a+b sec)	declined	89,817	4,347
cos^2 (A+C sec^2)/(a+b sec)	declined	89,330	3,999
cos^3 (B sec+C sec^2)/(a+b sec)	declined	296,561	4,349
cos^4 (B sec+C sec^2)/(a+b sec)	declined	609,741	5,579
cos^2 (A+B sec+C sec^2)/(a+b sec)	declined	90,482	12,065
sin(x)^2/(a+b*csc(x))	declined	2,159	898

Rubi corpus, 5 s budget, whole corpus plus a 14,707-problem pocket:

• Wrong answers: 0.
• Gains: 4.5.1.2:571, 4.5.3.1:384 and 4.5.4.2:816/962/963/1096. They time out on master and solve here in 0.1 s.
• Master's one row: 4.7.7:117 takes master 22 s, so it lands at either side of the budget.
• Unit suite: 15,230 passed.
• Allocation gate: passed.

A test differentiates five of these answers back at pinned values and bounds their sizes. The breaking-changes entry is under "A power of the cosine beside a secant binomial is integrated in the cosine".

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura