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The radicand an inverse sine or cosine holds is read as one - #1489
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arcsin(u) holds the radicand 1 - u^2 without writing it: its derivative is u'/sqrt(1 - u^2). Beside arcsin(c x), the factors d + c d x and f - c f x multiply to d f (1 - c^2 x^2), a multiple of that radicand. The rule that writes such a multiple over the radicand (#1487, #1488) took its radicands only from logarithms, so the pair ran out of time. The radicand of an inverse sine or cosine of a linear is now one of the radicands it reads. Two factors whose product is the radicand itself, sqrt(1 + c x) sqrt(1 - c x), are written as its one root too. Rubi's 5.1.4, 5.1.5, 5.2.4 and 5.2.5, all 504 problems that count: 405 to 463, no row lost, 77 timeouts to 19. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_012sonx8iAspMiwRwokT1Ura
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(d + c d x)^(1/2) (f - c f x)^(3/2) (a + b arcsin(c x))ran out of time.arcsin(u)holds the radicand1 - u^2without writing it, since its derivative isu'/sqrt(1 - u^2). The two factors besidearcsin(c x)multiply tod f (1 - c^2 x^2), which is the case #1487 and #1488 answer. Those rules took their radicands only from logarithms.The radicand of an inverse sine or cosine of a linear is now one of the radicands the rule reads. Two factors whose product is the radicand itself,
sqrt(1 + c x) sqrt(1 - c x)besidearccos(c x), are written as its one root too.One answer changes form on master.
(d - c^2 d x^2)^(3/2) (a + b arcsin(c x))was answered withd^(3/2)in front, from #1481's constant pull. It now getsK = sqrt(d - c^2 d x^2)/sqrt(1 - c^2 x^2), which is Rubi's form and equalsd^(1/2)whereverarcsin(c x)is real. Both were unevaluated in 2.5.0, which is what the changelog entry compares with.Measured
(d + c d x)^(1/2) (f - c f x)^(3/2) (a + b arcsin(c x))The new theory checks five shapes by differentiating back at five points inside
|2x| < 1, with numeric coefficients.Found on the way, and not changed here: with
MathS.Settings.DowncastingEnabledoff,(a + b arcsin(c x))/sqrt(d - c^2 d x^2)integrates toNaNon master and on this branch. With the default setting it is answered correctly. Filed as #1490.Part of #718.
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