The other four trigonometric functions are written as sine and cosine before the complement of a substitution is, and never where that leaves an odd power of the complement under a root

[Rafael-SOWNet]

Charlwood's arctan(sqrt(sec(x) - 1)) sin(x) had no antiderivative. By parts it leaves
sin(x)/sqrt(sec(x) - 1), and under u = cos(x) -- the complement of the written
sine, a candidate since #1307 -- that is -1/sqrt(1/u - 1), a root of a quotient of
two linears, which is the linear-radical substitution's since #1308. With the
secant left standing it kept its x and the candidate was refused. The pass
that writes the complement's even powers in u writes the secant, cosecant,
tangent and cotangent of the argument as quotients of the sine and cosine
first now, and substitutes u again where that has written the function anew.

And refuses the rewriting where it leaves an odd power of the complement under
a root. The first draft did not, and Timofeev's
(sec(x)^2 - 3 sqrt(4 sec(x)^2 + 5 tan(x)^2) tan(x))/(sin(x)^2 (...)^(3/2)) under
u = sin(x) became a root of a quotient by cos(x)^2, which the simplification the
substitution runs takes as .../cos(x) -- its value where the cosine is positive
and its negative elsewhere -- and the integrand was answered wrongly on half the
line, measured at two of six sampled points and pinned as declined-or-right on
both sides. An even power of the complement under a root is 1 - u^2 by then and
is no such hazard.

One answer changes shape: sin(x) sin(2x) over cos(x) is 2 sin(x)^2 once the
simplification has written the double angle out, and the second pass reads the
sine that wrote as u again, so it is 2 sin(x)^3/3 now rather than
product-to-sum's cosines of x + 2x; the test that pinned the old shape pins
the new one.

Measured

Every answer differentiated back on both sides of zero.

Rubi corpus, 463-problem sample, on the same evening:

master at #1309     401/463, 0 wrong, 0 timeouts, 100 s
with this           402/463, 0 wrong, 0 timeouts, 99 s

One more, nothing lost: Charlwood 32.

The five test suites are green.

Part of #718.

🤖 Generated with Claude Code

https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura