Powers of a + a sec and c - c sec that add to a negative whole number are integrated in the first sum

[Rafael-SOWNet]

Part of #718.

sec(e + f x) (a + a sec(e + f x))^m (c - c sec(e + f x))^(-1 - m) and two more of Rubi's 4.5.2.3 (#189–#191) were declined. Here the two powers add to a negative whole number -k, not to a half-odd number, which is what #1829 handles. In u = a + a sec, the conjugate a - a sec is 2a - u, and the integral is

expr / u' · (2a)^(-k) · Σ_{j<k} C(k-1, j) u^(j+1) (a - a sec)^(k-j) / (m + j + 1/2)

where expr is the integrand and m is the first power. For k = 1 that is -tan(y) (a + a sec)^m (c - c sec)^(-1-m) / (2m + 1). The cosecant's integral is the same.

Limits:

• k goes up to 12.
• The power sum is decided with Simplify, because InnerSimplified leaves m + -1 - m as it is.
• The answer is returned only if it differentiates back at sampled points.

Measured on 99fbaa7 (master) and on 2.5.0:

Integrand	2.5.0	master 99fbaa7	this branch
sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-1-m)	declined	declined, 3.4 s	245 characters, 0.3 s
sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-2-m)	declined	declined, 2.3 s	324
sec(e+f*x)*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))^(-3-m)	declined	declined, 3.3 s	402
sec(x)*(1+sec(x))^m*(1-sec(x))^(-1-m)	declined	declined	123
sec(x)*(2+2*sec(x))^(5/3)*(3-3*sec(x))^(-8/3)	declined	declined	168
sec(x)*(1-sec(x))^m*(1+sec(x))^(-1-m)	declined	declined	124
csc(x)*(2+2*csc(x))^m*(3-3*csc(x))^(-2-m)	declined	declined, 1.2 s	233

ConjugateSecantSumsWhosePowersAddToANegativeWholeTest differentiates each answer back at three points after substituting m, a, c, e and f. The corpus marks #189–#191 Unverified, not Solved, because the integrand is complex at its real sample points, as it was for #1829.

Corpus chain against master a4993e6:

• The 4.5.2.3 pocket (on its own first) plus families 0–8: three moves, Entity parent #189–FastExpression into linq expression #191 gained; 0 wrong answers; no new timeouts.
• Allocation gate: passed.

Before merging 99fbaa7, the suite run inside the chain was killed by the memory guard at 8 GB after 9,117 tests, none failing. After the merge, the whole suite run on its own passed 15,277. The every-target build succeeds with 0 warnings.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura