bug report: integrate(acos(x^2), x) returns incorrect antiderivative (FriCAS 1.3.12)

[Fabian]

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

[Dima Pasechnik]

I can add that I can repeat this at Fricas prompt, it has nothing to
do with Sage per se.
It's for me on Gentoo Linux, with

$ fricas
Checking for foreign routines
FRICAS="/usr/lib64/fricas/target/x86_64-pc-linux-gnu"
spad-lib="/usr/lib64/fricas/target/x86_64-pc-linux-gnu/lib/libspad.so"
foreign routines found
openServer result 0
FriCAS Computer Algebra System
Version: FriCAS 1.3.12 built with sbcl 2.5.11
Timestamp: Thu Dec 11 00:10:42 CST 2025

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[Fabian]

Hello Dima,

This was a fast reply!

Fabian

[Kurt Pagani]

The integral cannot be expressed in terms of elementary functions. When you take 'complexIntegrate' it should work, although Google KI says that it requires the Hypergeometric function [2]F[1] to evaluate the second integral "integrate(x^{^2}/sqrt(1-x^{^4}),x)" which is left after partial integration, so *no* guarantees if (1) below is correct 😉 However, I think it is ...

To verify the result below, one has also to use the
fact that

acos(x) = -%i * log(x+sqrt(x^{^2}-1)) holds.


(1) -> complexIntegrate(acos(x^{^2}),x)

   (1)
                      +------+
                      | 4         2             +------+
        2 +---+    - \|x  - 1  - x        +---+ | 4
       x \|- 1 log(----------------) - 4 \|- 1 \|x  - 1
                     +------+
                     | 4         2
                    \|x  - 1  - x
     +
           +---+          1            +---+          1
       4 x\|- 1 ellipticF(-,- 1) - 4 x\|- 1 ellipticE(-,- 1)
                          x                           x
  /
     2 x
                                                    Type: Expression(Integer)

[Qian Yun]

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.

- Best,
- Qian

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[Waldek Hebisch]

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.

> To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel...@googlegroups.com.
> To view this discussion visit https://groups.google.com/d/msgid/fricas-devel/e34f4198-37cd-463a-9c65-fba93aa7f97f%40gmail.com.

--
Waldek Hebisch

[Fabian]

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

[Kurt Pagani]

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).

[Fabian]

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

[Kurt Pagani]

On Wednesday, 28 January 2026 at 14:10:53 UTC+1 Fabian wrote:

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.

This might explain it, however the culprit must be 'eval' since the lines below ~prove that D(F,x)=f=acos(x^2)  holds.

                       FriCAS Computer Algebra System
         Version: FriCAS 2025.12.23git built with sbcl 2.2.9.debian
                   Timestamp: Fri  9 Jan 20:25:59 CET 2026
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------

   Function declaration sixel : TexFormat -> Void has been added to
      workspace.
   Function declaration lisp : String -> SExpression has been added to
      workspace.
f:=acos(x^2);


                                                    Type: Expression(Integer)
F:=complexIntegrate(f,x);


                                                    Type: Expression(Integer)
R:=complexNormalize(D(F,x)-f);


                                                    Type: Expression(Integer)

real R --> 0


   (14)  0
                                                    Type: Expression(Integer)

c:=numer imag R ;


      Type: SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))

subst(c, sqrt(x^4-1)=sqrt(x^2+1)*sqrt(x^2-1) ) --> 0


   (16)  0
                                                    Type: Expression(Integer)
(17) ->

 

Fabian

[Waldek Hebisch]

On Wed, Jan 28, 2026 at 05:10:53AM -0800, Fabian wrote:
> 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.

Note that in symbolic computation branches have "equal rights".
More precisly, Galois theory say that branches are indistinguishable
in algebraic way. That is numeric computation which makes choice
of branches.

One, essentially unavoidable trouble here is definiton of elliptic
integrals: they are defined using product of roots, while algebraic
function needed to complete your integral has a single root:

(47) -> acos(x^2) - D(x*acos(x^2), x)

2
2 x
(47) -----------
+--------+
| 4
\|- x + 1
Type: Expression(Integer)
(49) -> D(-2*ellipticF(x, -1) + 2*ellipticE(x, -1), x)

2
2 x
(49) --------------------
+--------+ +------+
| 2 | 2
\|- x + 1 \|x + 1
Type: Expression(Integer)

Different trouble is FriCAS generates another expression for integral,
which needs complex numbers for real x in (-1, 1) and numeric
evaluation chooses wrong branches. This in general is unsolvable
problem, but in simple cases like this FriCAS should produce version
above and not the complex one.

--
Waldek Hebisch