Kurt: complexNumeric eval(J,x=1/2) also gives approximately 0, when I
include your code in a Sage code and run it in the SageMathCell. With
I:=complexIntegrate(acos(x^2),x) the following line gives almost 0:
complexNumeric eval(D(I,x)+acos(x^2),x=1/2)
This suggests that D(I,x)=-acos(x^2) . This differs from the correct
expression acos(x^2) by a sign. I suspect that the mistake comes from
choosing wrong branches of the logarithm and square root function.
Fabian
On Wednesday, January 28, 2026 at 12:04:23 PM UTC+1 Kurt Pagani wrote:
> This is strange because I get:
>
> I:=complexIntegrate(acos(x^2),x);
> J:=complexNormalize(D(I,x)-acos(x^2));
> complexNumeric eval(J,x=1/2) --> .. E-20
>
> tst(z) == complexNumericIfCan eval(J,x=z) --> ~0 for most z ;)
>
> There must be another issue (eval doesn't commute wiht complexNormalize):
>
> complexNumeric complexNormalize(eval(D(I,x)-acos(x^2),x=1/2)) --> -
> 2.6362321433 - 0.7 E -20 %i
>
> complexNumeric eval(complexNormalize(D(I,x)-acos(x^2)),x=1/2) --> -
> 0.1866265314 E -20 - 0.7228014483 E -20 %i
>
> By the way, the result of complexIntegrate(acos(x^2),x) seems to be
> correct, at least in fricas.
>
> Using the log repr of acos, D(I,x)-acos(x^2) reads:
>
> R:=normalize (D(I,x)+%i*log(x^2+%i*sqrt(1-x^4)));
> real R --> 0
> imag numer R --> 0 (after a manual subst).
>
>
> On Wednesday, 28 January 2026 at 09:11:26 UTC+1 Fabian wrote:
>
>> Hello FriCAS group,
>>
>> Thank you Kurt, Qian, and Waldek for your prompt replies. I have tried
>> out the suggestion by Kurt to use complexIntegrate. The following check
>> indicates that this does not work for acos(x^2):
>>
>> from sage.interfaces.fricas import fricas
>> fricas.eval(
>> "f:=acos(x^2);"
>> "F:=complexIntegrate(f,x);"
>> )
>> print(fricas("complexNumeric(eval(D(F,x)-f,x=1/2))"))
>>
>> This produces the output - 2.63..., which is not close to desired value 0.
>>
>> Fabian
>>
>> On Wednesday, January 28, 2026 at 3:33:46 AM UTC+1 Waldek Hebisch wrote:
>>
>>> On Wed, Jan 28, 2026 at 10:09:14AM +0800, Qian Yun wrote:
>>> > Thanks for the report.
>>> >
>>> > Git bisect points to
>>> >
>>> https://github.com/fricas/fricas/commit/1f42999f91ce516a8d027a61be4ecbf32ad2ada4
>>>
>>> >
>>> > "Handle some elliptic integrals", June 14, 2022.
>>> > (Between 1.3.7 and 1.3.8)
>>> >
>>> > Before this commit, the result is a integral sign
>>> > which means fricas proves it does not have elemental
>>> > integral, which is correct.
>>>
>>> The reason is that transformations used in postprocessing result
>>> of integration may incorrectly transform elliptic integrals to
>>> 0.
>>>
>>> >
>>> > - Best,
>>> > - Qian
>>> >
>>> > On 1/28/26 1:42 AM, Fabian wrote:
>>> > > Hello FriCAS group,
>>> > >
>>> > > It looks like I found a bug in FriCAS:
>>> > > F:=integrate(acos(x^2),x)
>>> > > gives:
>>> > > +--------+
>>> > > | 4 2 2
>>> > > - 2 \|- x + 1 + x acos(x )
>>> > > ----------------------------
>>> > > x
>>> > >
>>> > > D(F,x)-acos(x^2)
>>> > > gives the following instead of 0:
>>> > > 2
>>> > > -------------
>>> > > +--------+
>>> > > 2 | 4
>>> > > x \|- x + 1
>>> > >
>>> > > I use FriCAS via https://sagecell.sagemath.org/ . Here is some
>>> Sage-code
>>> > > that produces the above output:
>>> > >
>>> > > from sage.interfaces.fricas import fricas
>>> > > fricas.eval(
>>> > > "F:=integrate(acos(x^2),x)"
>>> > > )
>>> > > print("F=\n",fricas("F"))
>>> > > fricas.eval("f:=D(F,x)-acos(x^2)")
>>> > > print("F'-acos(x^2)=\n",fricas("f"))
>>> > >
>>> > > The FriCAS version is 1.3.12. (print(fricas.eval(")lisp |
>>> > > $build_version|") tells me this.)
>>> > >
>>> > > The SageMath version is 10.8, Release Date: 2025-12-18. (version()
>>> tells
>>> > > me this.)
>>> > >
>>> > > Fabian
>>> > >
>>> > > --
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>>> send
>>> > > an email to [email protected] <mailto:fricas-
>>> > > [email protected]>.
>>> > > To view this discussion visit
>>> https://groups.google.com/d/msgid/fricas-
>>> > > devel/e5fde980-d9d7-4e5d-9fb9-d1cc5c45c021n%40googlegroups.com
>>> <https://
>>> > > groups.google.com/d/msgid/fricas-devel/e5fde980-d9d7-4e5d-9fb9-
>>> > > d1cc5c45c021n%40googlegroups.com?utm_medium=email&utm_source=footer>.
>>>
>>> >
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>>>
>>>
>>> --
>>> Waldek Hebisch
>>>
>>
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