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Integrate fractional trigonometric powers two apart by the reduction, and a sine raised from nothing as a derivative - #2031
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… and a sine raised from nothing as a derivative - A sum of powers of one sine, cosine, secant or cosecant of a linear, the powers not whole and two apart, by the reduction formula from the highest down; each power alone is elliptic. Beside x, by parts once. - The derivative of a sine raised from nothing times powers of the cosine and sums in it, or of a cosine beside the sine. - A cosine beside a function of the sine with a power of it that is not rational, in s = sin(x). - (c sin(a + b x)^n)^(1/n) as a constant times sin(a + b x) between the zeros of the sine, and so for the cosine. - Under the homogeneous substitution, factors written alike on both sides cancelled; under the tangent, a whole power of a product distributed; and the root of 1 - cos(x)^2 beside a root of 1 - cos(x) as the product of the roots of 1 - cos(x) and 1 + cos(x). Chain against master f7f2a7a: pocket of 2,005 rows, 1,979 -> 1,998 solved, 0 wrong; family 4's sample 1,025 -> 1,026, the others unchanged; the 19 rows that moved, alone, 0 -> 19 solved; 7,510 rows compared for allocation, none grew past 55 MB. The new test that the power-of-x case is left to the rules before it pinned `c > 0` in the answer to (c x^n)^(1/n), as those wrote it before #2012; master writes x (c x^n)^(1/n)/2, whose derivative is the integrand on every branch since d/dx (c x^n)^(1/n) = (c x^n)^(1/n)/x, and the test pins that. Part of #233. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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Part of #233.
Nineteen of Rubi's trigonometric rows that master f7f2a7a declines or runs out of time on:
(3 + 3 sec(c + d x)^2)/sqrt(sec(c + d x))was two declined integrals. Beside x, by parts once.(a + a cos(x))^(3/2)/cos(x)^(5/4).s = sin(x):cos(x)/(sin(x) + sin(x)^sqrt(2)).(c sin(a + b x)^n)^(1/n)as a constant timessin(a + b x)between the zeros of the sine, and the same for the cosine.sqrt(1 - cos(x)^2)beside a root of1 - cos(x)is written as the product of the roots of1 - cos(x)and1 + cos(x).Chain against master f7f2a7a:
The full unit suite passes locally. The one failure it had was this branch's own new test, which pinned
c > 0in the answer to(c x^n)^(1/n). Since #2012, master writes that asx (c x^n)^(1/n)/2, which is right on every branch, and the test now pins that. The BREAKING-CHANGES lengths were re-measured on this branch: five had moved with the rebase. The v2.5.0 column was measured on a v2.5.0 build.This overlaps with #2026 on
sqrt(1 - cos(x))/sqrt(a - cos(x)): both answer it. Whichever merges second gets its entry re-measured.🤖 Generated with Claude Code
https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura