Still nothing official is to be found on the internet of Masser and
Zannier's work that lead to counterexamples to a theorem by James H. Davenport: Some parametrized algebraic functions not identically
integrable can actually be integrated for infinitely many parameter
values.
Nothing official, that is, apart from David Masser's feature article "Integration in elementary terms" in the LMS Newsletter (Newsletter
London Math. Soc. 473 (2017), 30-36), made available here:
<https://webusers.imj-prg.fr/~jan.nekovar/co/ter/int.pdf>
[...]
They published it. https://www.intlpress.com/site/pub/pages/journals/items/acta/content/vols/0225/0002/a002/
To dev of FriCAS: apparently FriCAS cannot handle the key
result of both papers, it causes infinite loop (or whatever):
integrate(x/(x^2-1/5-2*%i/5)/(x^3-x)^(1/2), x)
While it is elementary! Mathematica 13 can do it very fast, but
not in elementary functions. It is interesting WHY FullSimplify
does not see it from the math. 13 result, is it possible there is
a simplification to elementary function possible or a constant
difference in real part? Or is the result in the paper too big?
__________
The other example that Mathematica 13 solves with insanely
big result. Never seen anything like it (but again paper gives
elementary result, DID not check FriCAS):
Integrate[((5t^2+40^t+62)x+t^3+8t^2+70^t+144)/ (x-t)((2t+8)x+t^2+4t+18)( x^3-30x-56)^(1/2),x]
P.S. After reading the papers I did not find the script to
generate those but it should be there, of course.
I suspect you are asking simply too much of Mathematica here.
In fact, we were able to show, partly computationally, that Q(i√2) does not turn up, and we
which Maple 18 cannot check even by differentiation (however it can check equality up to say1000 decimal places when we integrate between say x = 2 and x = 2.1).
Wait a second, it is you who is Detmar Martin Welz in the paper.
Then do you know where is this code? "Partly computationally" 138 t
of 21.12, what are they?
In fact, we were able to show, partly computationally, that Q(i√2)
does not turn up, and we
Oh and also I do not have the old preprint with wrong proofs, can you
give it?
BTW, the "sceptical" part is just nice, can you maybe also check what
did you wrote to them?
And what did you do with FriCAS?
There is also a numerical integration "proof", which they obviously
copied from 2017 paper:
which Maple 18 cannot check even by differentiation (however it can
check equality up to say 1000 decimal places when we integrate
between say x = 2 and x = 2.1).
I think that is illegal. You cannot just check small part from 2 to
21/10. Did not do FullSimplify, too lazy :), the other Maple 18
"comment" can be now done thanks to IntegrateAlgebraic:
SetSystemOptions[
"IntegrateOptions" -> {"IntegrateAlgebraicTimeConstraint" -> 100}];
Integrate[(5 x - 1)/Sqrt[x^4 + 2 x^2 - 4 x + 1], x]
It would nice of them to prove like here https://hdl.handle.net/2346/45299 Different kind of paper, I suppose.
I can confirm that the current online version of FriCAS
The integral on p.232 of the paper:
What is version you have in your linux? I use the latest, since
I use Debian testing. Will try 10 hours after all. Maupybe will
compile first HEAD of master.
I did check 1.3.7 in my debian, it also timeouts after 8 hours
and maybe even crashes, since it exits as if )quit happened.
Using division polynomials I have found points of order 5,
one corresponds to u^2 = (1 + 2*sqrt(-1))/5. FriCAS have
now computed integrals for orders 3, 5, 6, 8. The results
are rather lengthy, so instead of posting them here I have
put them at:
http://www.math.uni.wroc.pl/~hebisch/fricas/p3
(3 above means order 3, replace 3 by 5, 6, 8 for higher order).
found points of order 5,
one corresponds to u^2 = (1 + 2*sqrt(-1))/5.
Well, apparently the author reads this indeed:
https://github.com/fricas/fricas/commit/1f42999f91ce516a8d027a61be4ecbf32ad2ada4
of course this is very strange:
testIntegrate("(5*x-1)/sqrt(x^4 + 2*x^2 - 4*x + 1)", "x", "alg")
since it is really Zolotarev case, in fact same stuff is in https://en.wikipedia.org/wiki/Risch_algorithm
From the paper (p. 230): testIntegrate("5*x^2/sqrt(x^6 +x)", "x", "alg"),
but it is not that important. (Also "Massetr and Zanier", typos.)
found points of order 5,
one corresponds to u^2 = (1 + 2*sqrt(-1))/5.
Yes, that is in the paper too, page 233. In fact t = (1/5(5-10i))^2
is indeed (1 + 2*sqrt(-1))/5.
Both those examples work even before the commit.
I did compile HEAD of master today, nothing helps our case though.
Will compile 1.3.2 and check it out. Will not do bisect though, hope
author will find the regression commit.
Case closed, you can do it all with
setSimplifyDenomsFlag(true)
integrate(x/((x^2 - ((1 + 2*sqrt(-1))/5))*sqrt(x^3 - x)), x)
Apparently it does not like %i, since that https://github.com/fricas/fricas/pull/92#issuecomment-1157581265
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