Timeline for Does there exist a complete implementation of the Risch algorithm?
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| Oct 18, 2020 at 11:21 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Oct 18, 2020 at 7:25 | comment | added | Martin Rubey | Just for the record: the very first integral in your answer, involving the two parameters, is done by the development version of FriCAS (in about 15 minutes on my computer), whereas the other branches are still unimplemented. | |
| Oct 18, 2020 at 3:35 | history | edited | Sam Blake | CC BY-SA 4.0 |
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| Oct 18, 2020 at 2:33 | history | edited | Sam Blake | CC BY-SA 4.0 |
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| Oct 17, 2020 at 13:58 | history | edited | Sam Blake | CC BY-SA 4.0 |
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| Oct 17, 2020 at 0:35 | comment | added | Sam Blake | ... the substitution $u=\frac{x^6-2}{x^4}$, reduces the integral to $$\int \frac{\left(x^6+4\right) \left(2 x^6+x^4-4\right) \sqrt{8 x^{12}+7 x^{10}-4 x^8-32 x^6-14 x^4+32}}{x^9 \left(x^6-2\right)} dx = \int \frac{(2 u+1) \sqrt{8 u^2+7 u-4}}{2 u} du.$$ | |
| Oct 17, 2020 at 0:34 | comment | added | Sam Blake |
@TimothyChow, in terms of your roadmap - here's an example which returns a long-standing error in AXIOM/FriCAS "Error detected within library code: integrate: implementation incomplete (residue poly has multiple non-linear factors)" integrate(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+7*x^10+8*x^12)^(1/2))/(x^9*(-2+x^6)),x) Maple computes this integral int(convert(((4+x^6)*(-4+x^4+2*x^6)*(32-14*x^4-32*x^6-4*x^8+7*x^10+8*x^12)^(1/2))/(x^9*(-2+x^6)),RootOf),x); and my heuristic quickly recognises that...
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| Oct 17, 2020 at 0:12 | comment | added | Sam Blake | @TimothyChow, one of the issues is even if a complete implementation was made, many integrals take an inordinate amount of time to compute. My heuristic method (referenced in the answer) quickly computes many integrals that take hours to return an answer in FriCAS/AXIOM. | |
| Oct 16, 2020 at 17:28 | comment | added | Dima Pasechnik | The problem on the implementation part is that Fricas is a very complex and somewhat esoteric system, the commitment required from a potential contributor is quite big. | |
| Oct 16, 2020 at 15:05 | comment | added | Martin Rubey | @TimothyChow I'm afraid the problem is that there is essentially one person working on this, which is Waldek Hebisch. He is brilliant, though. | |
| Oct 16, 2020 at 14:39 | comment | added | Timothy Chow | Extremely informative response...thanks! I am wondering if there is some kind of "roadmap" that delineates the main challenges that stand in the way of a complete implementation. The Fricas "Risch Implementation Status" page mentioned by Dima Pasechnik is a start but is vague about some important details. If there were a clearly mapped-out plan of attack, perhaps it would attract more researchers to the area, and the challenges could be knocked off one by one. | |
| Oct 16, 2020 at 13:09 | comment | added | Martin Rubey | It might be interesting to re-run your testsuite. If this was a bug, then possibly it affected also other examples. | |
| Oct 16, 2020 at 12:53 | comment | added | Sam Blake | @MartinRubey Looks like the dev version of FriCAS has fixed this example. I ran it on 1.3.6. | |
| Oct 16, 2020 at 9:15 | comment | added | Martin Rubey | I cannot verify you claim that fricas returns the integral above unevaluated. I get: ${-{x \ {\log \left( {{{-{2 \ {\sqrt {{{{x} ^ {8}}+{{x} ^ {5}}+{2 \ {{x} ^ {4}}}+x+1}}}}+{2 \ {{x} ^ {4}}}+x+2} \over x}} \right)}} -{x \ {\sqrt {3}} \ {\arctan \left( {{{{\left( {2 \ {{x} ^ {4}}} -x+2 \right)} \ {\sqrt {3}}} \over {6 \ {\sqrt {{{{x} ^ {8}}+{{x} ^ {5}}+{2 \ {{x} ^ {4}}}+x+1}}}}}} \right)}}+{4 \ {\sqrt {{{{x} ^ {8}}+{{x} ^ {5}}+{2 \ {{ x} ^ {4}}}+x+1}}}}} \over {{16} \ x} $ within a second, at least with the current version from git. | |
| Oct 16, 2020 at 1:02 | comment | added | Sam Blake | To be fair, here's one which may be computed with FriCAS and AXIOM, but returns an error in Maple: int(convert((x^3+1)*(x^6-x^3-2)^(1/2)/x^4/(x^6-2*x^3-1),RootOf),x); | |
| Oct 16, 2020 at 0:51 | comment | added | Sam Blake | @MartinRubey, I didn't mean to imply that AXIOM and/or FriCAS could not compute this integral. Here's an example which Maple and AXIOM can compute and FriCAS claims is not elementary: int(convert(((-1+3*x^4)*(1+x+2*x^4+x^5+x^8)^(1/2))/(x^2*(4+x+4*x^4)),RootOf),x); | |
| Oct 15, 2020 at 13:22 | history | edited | Sam Blake | CC BY-SA 4.0 |
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| Oct 15, 2020 at 12:42 | comment | added | Martin Rubey | actually, fricas does this integral in a second | |
| Oct 15, 2020 at 12:31 | history | edited | Timothy Chow | CC BY-SA 4.0 |
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| Oct 15, 2020 at 12:29 | vote | accept | Timothy Chow | ||
| Oct 15, 2020 at 1:48 | review | First posts | |||
| Oct 15, 2020 at 2:01 | |||||
| Oct 15, 2020 at 1:43 | history | answered | Sam Blake | CC BY-SA 4.0 |