The arcsecant beside a root of x^2 - 1 is integrated right below -1 too

[Rafael-SOWNet]

Part of #718.

Under x = sec(u), the substitution that undoes the arcsecant, sqrt(x^2 - 1) is |tan(u)|, and on the arcsecant's range the tangent has the sign of x. It was written tan(u), so these were answered right for x > 1 and wrong for every x < -1, where they are as real -- the derivative of the answer was the integrand's negative there:

integrand	master 51cb049a	this
x arcsec(x)/sqrt(x^2 - 1)	wrong for x < -1	right on both intervals
arcsec(x)/(x^2 sqrt(x^2 - 1))	wrong for x < -1	right on both intervals
arcsec(x)^2/(x sqrt(x^2 - 1))	wrong for x < -1	right on both intervals
arcsec(x + 3)^2/((x + 3) sqrt((x + 3)^2 - 1))	wrong for x < -4	right on both intervals

Checked with the library's own settings, the first three at x = -2.7, -1.9, -1.3, 1.3, 2.6 and the fourth at x = -6, -4.5, -1.5, -0.5, 0.5, 1.5: on master the derivative is the integrand's negative at every point below -1 for the first three and below -4 for the fourth, and right at the rest; here it is right at all of them.

What changes. The secant's root carries a sign, tan(u) sgn, as the cosecant's and the cotangent's already did, and a first power of the arcsecant beside it is left to parts as theirs is. The sign is written back as the argument's. The other two wrote theirs back as the sign of x, which is the argument's only for a positive slope and no constant: arcsec(x + 3) has the sign of x + 3, and at x = -1.5, where the fourth row is checked, the two differ. On 2.5.0 these were declined.

Tests: InverseSecantIntegralTest.BesideARootOnBothIntervals, nine rows -- Timofeev's, Charlwood's and the linear argument -- each differentiated back on both intervals. All nine fail on master.

Measured first with work/intbench's IB_BOTHSIDES=1, which also compares at the negatives of its sample points, on the 314 problems with an arcsecant, an arccosecant or an arccotangent in them, against master 8f3757cd, the branch's base:

	master	this
solved	291	301
wrong	8	0

The eight are the arcsecant beside its root, Timofeev's and Charlwood's.

Measured then on the Rubi corpus against master 8f3757cd:

	master	this
family 0, independent suites (1814)	1766	1766
family 1, 40 a file (1381)	1296	1296
families 2 to 8, sampled (2410)	2215	2219

No answer is wrong in the sample on either build. Of the eight problems the two builds disagreed on, run again one build at a time, three ran out of time on master and are answered here -- 5.5.1's (a + b asec(c x))/(d + e x)^2 and its cube, and one of 6.6.1's -- and one went the other way: sqrt(g sin(e + f x))/((c + d sin(e + f x)) sqrt(a + a sin(e + f x))), which has no arcsecant in it and takes 23 to 35 s on either build without a budget, at the edge of the corpus's. The other four are answered on both.

The suite passes on the commit measured, 14893b8a, 14,600 tests, and so does the allocation gate. Master 51cb049a is merged in since, without conflicts; the calculus tests pass on the merge, 3,992 of them, and the four rows above are right at every point on it.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura