[Rafael-SOWNet]

Follows #1959, which answered an even number of roots of a constant times the secant or cosine on both sides of the cosine's zeros. This does the same for an odd number.

For N roots, n of them of the cosine, let s be the sign of the product of their constants. Where the cosine is negative:

• the turns multiply to (-1)^(N+n) s;
• the integrand is real there exactly where (-1)^N s > 0;
• where it is real, the turns are (-1)^n.

For an odd N the root of 1 - t^2 stays in the answer in t. That answer is kept beyond t^2 < 1 only where its derivative holds there at sampled points of both signs. It then carries the cosine's sign in front where n is odd, and holds on the whole real domain. Where s is not known, the condition is now provided <product> < 0 or cos(y) >= 0. The cosecant takes the same path through the complement, with sgn(sin(y)).

sqrt(-1 - sec(x)) sqrt(sec(x)) sqrt(cos(x)) is real only where the cosine is negative, and its answer had no value anywhere the integrand is real. It holds there now. 2.5.0 declined these integrands.

No Rubi row has an odd root count above one. Measured with the canonical intbench (its Partial verdict included) on every fair row of Rubi's 4.5.x and 4.6.x, 2,966 rows, master a639f292 against the branch:

• 0 Partial on both builds;
• no row moved to Partial or Wrong;
• the answers on all 1,317 rows with roots are byte-identical.

One row moved from Solved to Timeout under load 20 (4.6.7:19). Re-run back to back, it is solved on both builds: 20.8 s on master, 18.2 s on the branch.

DerivativeHoldsAtSampledPoints gains an optional set of points. OnePlusASecantHalfPowerIntegralTest adds 18 cases at both signs of the cosine and of the constants, compared wherever the integrand is real. They replace a case that asserted the old condition. BREAKING-CHANGES.md has the entry, and the half-angle entry's sentence on three roots is corrected.

Part of #233.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura