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we need more integral solvers #233
Description
There are a lot of stuff to do, but to begin with, I will put some unsolvable by AM integrals here:
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cos(x^2)*xvariable substitution method, WolframAlpha -
sin(x)*e^xintegrating by parts and solving integral equation, WolframAlpha -
1/(a^2 + x^2)standard integral for arctan, WolframAlpha -
x^2 / (x^4 + 1)partial fractioning WolframAlpha -
sqrt(tan(x))very painful, requires different solvers, WolframAlpha
I do not mention special integrals here, like e^(x^2) or 1/ln(x)
Activity
Add five families of standard integrals (#233) (#675)
Integrate a quotient whose numerator is not constant (#233) (#681)
Split a rational function at a rational root of its denominator (#233) (
Integrate a constant over x squared times the root of a quadratic (#691)
Rafael-SOWNet commentedon Aug 16, 2026
Three of the five are done. Measured on
masterata45a7256, each verified by differentiating the answer back rather than by eye.cos(x^2)*xsin(x^2) / 2 + Csin(x)*e^xe^x * (sin(x) - cos(x)) / 2 + C1/(a^2 + x^2)piecewiseon the sign ofa^2, witharctan(x / sqrt(a^2)) / sqrt(a^2) + Con the positive branchx^2 / (x^4 + 1)sqrt(tan(x))So the substitution case, the by-parts-and-solve-the-equation case, and the arctan standard integral are all answered. The two left are the two the body itself flags as hardest — partial fractioning over an irreducible quartic, and
sqrt(tan(x)), which it calls "very painful, requires different solvers". Both are also the twocasbenchreports unsolved, so this agrees with the corpus.On the arctan one, a correction to my own first reading. Differentiating it back and simplifying does not reduce to zero, and I nearly reported that as a wrong answer. It is not — the
piecewisecarries branch conditions that block symbolic cancellation. Checked numerically instead:1/51/501/131/1304/54/501/411/410Worth recording because it will catch the next person too: an antiderivative that comes back as a
Piecewisecannot be verified by simplifyingd/dx F - fto zero, and a harness that does so will report a false defect.Suggest ticking the three and leaving this open for the remaining two.
A power of the variable is a substitution even where it is written no…
A biquadratic denominator is decomposed over the reals (#1145)
A fractional power, and the tangent, are substitutions (#1146)
7 remaining items
Children taken up now pull their parent forward
A parent is never due before its children (#1626)