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 I 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)
On 27/01/2026 19:25, Fabian wrote:
Hello Dima,
This was a fast reply!
Fabian
On Tuesday, January 27, 2026 at 6:52:13 PM UTC+1 Dima Pasechnik wrote:
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
On Tue, Jan 27, 2026 at 11:42 AM Fabian <[email protected]> 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/ <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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