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A root the substitution search would write through i is kept, and the cube is answered too - #1552
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Under a candidate `u`, the substitution search simplifies the rewritten integrand one level, and that writes `sqrt(-u)` as `i sqrt(u)`: the other root wherever u is negative. `sqrt(a + b sech(x)) tanh(x)^5` meets it with `u = tanh(x)^2 - 1`, negative everywhere, and was answered through `i`, off at every real point, and then declined by the tangent substitution's check. A candidate whose integrand holds the imaginary unit where the integrand did not is now passed over, and the search goes on to one under which the root stays real. The cube, `sqrt(a + b sech(x)) tanh(x)^3`, is still declined, as on master, and is left for a follow-up. Measured against master 40b7e58, together with the other two fixes opened beside it: a sample of families 1 to 7 (912 problems) 764 solved on both, row for row, and family 0 1750/1814 on both, 0 wrong. Part of #1370. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
This was referenced Sep 28, 2026
…it through i The cube, `sqrt(a + b sech(x)) tanh(x)^3`, was still declined. Under `u = tanh(x)^2 - 1` the one-level simplification of its integrand in u went one step further than the fifth power's, from `sqrt(-u)` to `i sqrt(u)`, so the candidate was passed over, where the fifth power's kept `sqrt(-u)` and was answered through it. Each one-level simplification of a candidate's integrand now keeps the integrand as written where the simplified one would hold the imaginary unit and neither the written one nor the integrand in x does. Both powers are answered and differentiate back. Closes #1370. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Closes #1370:
sqrt(a + b sech(x)) tanh(x)^5andsqrt(a + b sech(x)) tanh(x)^3are both answered now, and right.The tangent substitution writes
sech(x)assqrt(1 - u^2)foru = tanh(x). The substitution search then takesu_2 = u^2 - 1and simplifies the rewritten integrand one level, and that writesb sqrt(-u_2)asi b sqrt(u_2). That is the other root whereveru_2is negative, andtanh(x)^2 - 1is negative everywhere. The fifth power was answered ini sqrt(u^2 - 1), and its derivative was off at every real point, until the tangent substitution's check turned it into a decline.Two commits:
i, at each of the four places the search simplifies one. Traced, the cube's simplification went one step further than the fifth power's, fromsqrt(-u_2)toi sqrt(u_2), so the candidate that answers it was passed over. With the root as written, the search inu_2answers the cube throughsqrt(-u_2), as it answers the fifth power. The first commit's check stays, for whichever step writesiotherwise.sqrt(a+b*sech(x))*tanh(x)^5sqrt(a + b sqrt(1 - tanh(x)^2)), written in exponentialssqrt(a+b*sech(x))*tanh(x)^3ExponentialSubstitutionIntegralTest.ARootOfTheSecantBesideAnOddPowerOfTheTangentIsAnswereddifferentiates both back at five points with the symbols pinned.Measured at 780f638, both commits, against master 40b7e58:
tanh(c + d x)besidesqrt(a + b sech(c + d x)), its reciprocal and its minus three-halves power, the fifth powers and the cubes. Re-run alone, they are this branch's and no other PR's. The files' total time went from 470 s to 416 s.🤖 Generated with Claude Code