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SandBoxFriCAS

last edited 1 year ago by test1

Edit detail for SandBoxFriCAS revision 45 of 62

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Editor: h-x Time: 2019/11/09 02:35:02 GMT+0
Note: Still no CAS knows if this one is elementary... Probably it is not, but a fun fact...

added:

From h-x Sat Nov 9 02:35:01 +0000 2019
From: h-x
Date: Sat, 09 Nov 2019 02:35:01 +0000
Subject: Still no CAS knows if this one is elementary... Probably it is not,
	but a fun fact...
Message-ID: <20191109023501+0000@fricas-wiki.math.uni.wroc.pl>

\begin{axiom}
integrate((x+sin(x))^(1/3),x)
\end{axiom}

This page is set to execute \begin{axiom}... \end{axiom}
commands using FriCAS.  See also FriCASIntegration.
fricas
)version

Value = "FriCAS 1.3.5 compiled at Thu Feb 21 18:22:14 UTC 2019"

Any comments added here use this version of FriCAS?.


... --meliusja,  Tue, 08 Apr 2008 10:38:33 -0700 reply
fricas
solve(s=v*t+a*t^2/2,t)
\begin{equation*}
\label{eq1}\left[{{{2 \  t \  v}+{a \ {{t}^{2}}}-{2 \  s}}= 0}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))

... --meliusja,  Tue, 08 Apr 2008 10:39:28 -0700 reply
fricas
radicalSolve(s=v*t+a*t^2/2,t)
\begin{equation*}
\label{eq2}\left[{t ={{-{\sqrt{{4 \ {{v}^{2}}}+{8 \  a \  s}}}-{2 \  v}}\over{2 \  a}}}, \:{t ={{{\sqrt{{4 \ {{v}^{2}}}+{8 \  a \  s}}}-{2 \  v}}\over{2 \  a}}}\right]?\end{equation*}
Type: List(Equation(Expression(Integer)))

... --meliusja,  Tue, 08 Apr 2008 10:40:28 -0700 reply
fricas
solve(v*t+a*t^2/2-s=0,t)
\begin{equation*}
\label{eq3}\left[{{{2 \  t \  v}+{a \ {{t}^{2}}}-{2 \  s}}= 0}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))

... --meliusja,  Tue, 08 Apr 2008 10:41:13 -0700 reply
fricas
solve(v*t+a*t/2-s=0,t)
\begin{equation*}
\label{eq4}\left[{t ={{2 \  s}\over{{2 \  v}+ a}}}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))

... --meliusja,  Tue, 08 Apr 2008 10:42:37 -0700 reply
fricas
solve(v*t+a*t^2/2-s=0,t)
\begin{equation*}
\label{eq5}\left[{{{2 \  t \  v}+{a \ {{t}^{2}}}-{2 \  s}}= 0}\right]?\end{equation*}
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
radicalSolve(p^3 - p + 1/10=0,p)
\begin{equation*}
\label{eq6}\begin{array}{@{}l}
\displaystyle
\left[{p ={{{{\left(-{3 \ {\sqrt{- 3}}}+ 3 \right)}\ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}- 2}\over{{\left({3 \ {\sqrt{- 3}}}+ 3 \right)}\ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}, \: \right.
\
\
\displaystyle
\left.{p ={{{{\left(-{3 \ {\sqrt{- 3}}}- 3 \right)}\ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}+ 2}\over{{\left({3 \ {\sqrt{- 3}}}- 3 \right)}\ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}, \: \right.
\
\
\displaystyle
\left.{p ={{{3 \ {{\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{37
3}}}}\over{{60}\ {\sqrt{3}}}}}^{2}}}+ 1}\over{3 \ {\root{3}\of{{-{3 \ {\sqrt{3}}}+{\sqrt{-{373}}}}\over{{60}\ {\sqrt{3}}}}}}}}\right] \end{array}
\end{equation*}
Type: List(Equation(Expression(Integer)))

fricas
R1 ==> Record(foo1: String, remLexs: List Integer)
Type: Void
fricas
R2 ==> Record(foo2: String, remLexs: List Integer)
Type: Void
fricas
r1: R1 := ["a", [1,2,3]]
\begin{equation*}
\label{eq7}\left[{foo 1 = \verb#"a"#}, \:{remLexs ={\left[ 1, \: 2, \: 3 \right]?}}\right]\end{equation*}
Type: Record(foo1: String,remLexs: List(Integer))
fricas
r2: R2 := ["b", [5]]
\begin{equation*}
\label{eq8}\left[{foo 2 = \verb#"b"#}, \:{remLexs ={\left[ 5 \right]?}}\right]\end{equation*}
Type: Record(foo2: String,remLexs: List(Integer))
fricas
r1.remLexs
\begin{equation*}
\label{eq9}\left[ 1, \: 2, \: 3 \right]?\end{equation*}
Type: List(Integer)
fricas
r2.remLexs
\begin{equation*}
\label{eq10}\left[ 5 \right]?\end{equation*}
Type: List(Integer)

... --jib jabber,  Fri, 07 Mar 2014 07:04:39 +0000 reply
fricas
)set output tex off
 
fricas
)set output algebra on

guessPRec [1, 0, 1, 1, 4, 10, 35, 120, 455, 1792, 7413, 31780, 140833, 641928, 3000361, 14338702, 69902535, 346939792, 1750071307, 8958993507, 46484716684, 244187539270, 1297395375129, 6965930587924]

   (13)
   [
     [
       f(n):
                 2                            2
             (- n  - 17 n - 72)f(n + 3) + (4 n  + 30 n + 44)f(n + 2)
           + 
                  2                               2
             (19 n  + 113 n + 150)f(n + 1) + (14 n  + 42 n + 28)f(n)
         = 
           0
       ,
      f(0) = 1, f(1) = 0, f(2) = 1]
     ]
Type: List(Expression(Integer))

... --panacea,  Fri, 21 Nov 2014 18:25:05 +0000 reply
fricas
guessRec([1,1,0,1,- 1,2,- 1,5,- 4,29,- 13,854,- 685])

                                           2
   (14)  [[f(n): f(n + 2) + f(n + 1) - f(n)  = 0, f(0) = 1, f(1) = 1]]
Type: List(Expression(Integer))

integrate(exp(x-x^2), x) --mariuszi28cg,  Mon, 25 Sep 2017 20:14:29 +0000 reply
fricas
integrate(exp(x-x^2), x)

           1
           -
           4    2 x - 1  +---+
         %e erf(-------)\|%pi
                   2
   (15)  ---------------------
                   2
Type: Union(Expression(Integer),...)

test 1+1 --aUser,  Thu, 01 Feb 2018 14:32:47 +0000 reply
fricas
1+1

   (16)  2
Type: PositiveInteger?

^ or **? --aUser,  Thu, 01 Feb 2018 14:34:22 +0000 reply
2^3-2**3
2^3 --aUser,  Thu, 01 Feb 2018 14:34:46 +0000 reply
2^3
^ or **? --aUser,  Thu, 01 Feb 2018 14:35:09 +0000 reply
fricas
2^3

   (17)  8
Type: PositiveInteger?

^ or **? --aUser,  Thu, 01 Feb 2018 14:35:49 +0000 reply
fricas
2**3

   There are no library operations named ** 
      Use HyperDoc Browse or issue
                                 )what op **
      to learn if there is any operation containing " ** " in its name.

   Cannot find a definition or applicable library operation named ** 
      with argument type(s) 
                               PositiveInteger
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

bug? --aUser,  Thu, 01 Feb 2018 14:37:03 +0000 reply
fricas
sqrt(2)^2-2

   (18)  0
Type: AlgebraicNumber?

bug?  --aUser,  Thu, 01 Feb 2018 14:38:36 +0000 reply
fricas
a:=sqrt(2)

          +-+
   (19)  \|2
Type: AlgebraicNumber?
fricas
b:=sqrt(a)

          +----+
          | +-+
   (20)  \|\|2
Type: AlgebraicNumber?
fricas
b^2-a

   (21)  0
Type: AlgebraicNumber?

bug? --aUser,  Thu, 01 Feb 2018 14:41:20 +0000 reply
fricas
f(a) == { local b=sqrt(a); b^2-a}

  Line   1: f(a) == { local b=sqrt(a); b^2-a}
           ........A........B
  Error  A: Missing mate.
  Error  B: syntax error at top level
  Error  B: Possibly missing a } 
   3 error(s) parsing

... --aUser,  Thu, 01 Feb 2018 14:41:44 +0000 reply
fricas
f(a) == { local b:=sqrt(a); b^2-a}

  Line   1: f(a) == { local b:=sqrt(a); b^2-a}
           ..........A
  Error  A: syntax error at top level
  Error  A: Improper syntax.
   2 error(s) parsing

[q,r]? type --aUser,  Thu, 01 Feb 2018 15:00:38 +0000 reply
fricas
[q,r]

   (22)  [q, r]
Type: List(OrderedVariableList?([q,r]))

... --aUser,  Thu, 01 Feb 2018 15:03:11 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)
Type: Void
fricas
f(2)
fricas
Compiling function f with type PositiveInteger -> AlgebraicNumber 

   (24)  0
Type: AlgebraicNumber?
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

   (25)  0
Type: AlgebraicNumber?

bug --aUser,  Thu, 01 Feb 2018 15:03:46 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(2)
fricas
Compiling function f with type PositiveInteger -> AlgebraicNumber 

   (27)  0
Type: AlgebraicNumber?
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

   (28)  0
Type: AlgebraicNumber?
fricas
f(z)
fricas
Compiling function f with type Variable(z) -> Expression(Integer) 

   (29)  0
Type: Expression(Integer)
fricas
f(%i)
fricas
Compiling function f with type Complex(Integer) -> Expression(
      Complex(Integer)) 

   (30)  0
Type: Expression(Complex(Integer))

)display properties --aUser,  Thu, 01 Feb 2018 15:08:53 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(2)
fricas
Compiling function f with type PositiveInteger -> AlgebraicNumber 

   (32)  0
Type: AlgebraicNumber?
fricas
c:=f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

   (33)  0
Type: AlgebraicNumber?
fricas
)display properties c

Properties of c :
   Value (has type AlgebraicNumber):  0

... --aUser,  Thu, 01 Feb 2018 15:21:08 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2 = a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void

... --aUser,  Thu, 01 Feb 2018 15:21:23 +0000 reply
\begin{AxiomCode?}
f(a) == ( local b; b:=sqrt(a); b^2-a)
f(2)
\end{AxiomCode?}
... --aUser,  Thu, 01 Feb 2018 15:22:39 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2 = a)

   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(2)
fricas
Compiling function f with type PositiveInteger -> Equation(
      AlgebraicNumber) 

   (36)  2 = 2
Type: Equation(AlgebraicNumber?)
fricas
f(%i)
fricas
Compiling function f with type Complex(Integer) -> Equation(
      Expression(Complex(Integer))) 

   (37)  %i = %i
Type: Equation(Expression(Complex(Integer)))
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> Equation(
      AlgebraicNumber) 

          +-+    +-+
   (38)  \|2  = \|2
Type: Equation(AlgebraicNumber?)
fricas
f(2^(1/2))

          +-+    +-+
   (39)  \|2  = \|2
Type: Equation(AlgebraicNumber?)
fricas
f(2^(1/3))

         3+-+   3+-+
   (40)  \|2  = \|2
Type: Equation(AlgebraicNumber?)

)set output --aUser,  Thu, 01 Feb 2018 15:29:41 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

   (42)  0
Type: AlgebraicNumber?
fricas
)set output fortran

--------------------------- The fortran Option ----------------------------

 Description: create output in FORTRAN format

 )set output fortran is used to tell FriCAS to turn FORTRAN-style output
printing on and off, and where to place the output.  By default, the
destination for the output is the screen but printing is turned off.

Also See: )set fortran

Syntax:   )set output fortran <arg>
    where arg can be one of
  on          turn FORTRAN printing on
  off         turn FORTRAN printing off (default state)
  console     send FORTRAN output to screen (default state)
  fp<.fe>     send FORTRAN output to file with file prefix fp and file
              extension .fe. If not given, .fe defaults to .sfort.

If you wish to send the output to a file, you must issue this command
twice: once with on and once with the file name. For example, to send
FORTRAN output to the file polymer.sfort, issue the two commands

  )set output fortran on
  )set output fortran polymer

The output is placed in the directory from which you invoked FriCAS or
the one you set with the )cd system command.
The current setting is:  Off:CONSOLE 
f(sqrt(2))

   (43)  0
Type: AlgebraicNumber?
fricas
)set output tex

----------------------------- The tex Option ------------------------------

 Description: create output in TeX style

 )set output tex is used to tell FriCAS to turn TeX-style output
printing on and off, and where to place the output.  By default, the
destination for the output is the screen but printing is turned off.

Syntax:   )set output tex <arg>
    where arg can be one of
  on          turn TeX printing on
  off         turn TeX printing off (default state)
  console     send TeX output to screen (default state)
  fp<.fe>     send TeX output to file with file prefix fp and file
              extension .fe. If not given, .fe defaults to .stex.

If you wish to send the output to a file, you must issue this command
twice: once with on and once with the file name. For example, to send
TeX output to the file polymer.stex, issue the two commands

  )set output tex on
  )set output tex polymer

The output is placed in the directory from which you invoked FriCAS or
the one you set with the )cd system command.
The current setting is:  Off:CONSOLE 
f(sqrt(2))

   (44)  0
Type: AlgebraicNumber?
fricas
)set output script

---------------------------- The script Option ----------------------------

 Description: display output in SCRIPT formula format

 )set output script is used to tell FriCAS to turn IBM Script formula-style
output printing on and off, and where to place the output.  By default, the
destination for the output is the screen but printing is turned off.

Syntax:   )set output script <arg>
    where arg can be one of
  on          turn IBM Script formula printing on
  off         turn IBM Script formula printing off (default state)
  console     send IBM Script formula output to screen (default state)
  fp<.fe>     send IBM Script formula output to file with file prefix fp
              and file extension .fe. If not given, .fe defaults to .sform.

If you wish to send the output to a file, you must issue this command
twice: once with on and once with the file name. For example, to send
IBM Script formula output to the file polymer.sform, issue the two commands

  )set output script on
  )set output script polymer

The output is placed in the directory from which you invoked FriCAS or
the one you set with the )cd system command.
The current setting is:  Off:CONSOLE 
f(sqrt(2))

   (45)  0
Type: AlgebraicNumber?

... --aUser,  Thu, 01 Feb 2018 15:30:46 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

   (47)  0
Type: AlgebraicNumber?
fricas
)set output fortran on

f(sqrt(2))

      R48=0

   (48)  0
Type: AlgebraicNumber?
fricas
)set output tex on 

f(sqrt(2))

      R49=0

   (49)  0
\begin{equation}
\label{eq11}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output script on

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R50=0

   (50)  0
\begin{equation}
\label{eq12}0\end{equation}
Type: AlgebraicNumber?

... --aUser,  Thu, 01 Feb 2018 15:31:59 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

.eq set blank @
:df.
0 
:edf.

      R52=0

   (52)  0
\begin{equation}
\label{eq13}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output fortran console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R53=0

   (53)  0
\begin{equation}
\label{eq14}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output tex console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R54=0

   (54)  0
\begin{equation}
\label{eq15}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output script console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R55=0

   (55)  0
\begin{equation}
\label{eq16}0\end{equation}
Type: AlgebraicNumber?

... --aUser,  Thu, 01 Feb 2018 15:33:05 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

.eq set blank @
:df.
0 
:edf.

      R57=0

   (57)  0
\begin{equation}
\label{eq17}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output fortran on
 
fricas
)set output fortran console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R58=0

   (58)  0
\begin{equation}
\label{eq18}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output tex on 
 
fricas
)set output tex console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R59=0

   (59)  0
\begin{equation}
\label{eq19}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output script on
 
fricas
)set output script console

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R60=0

   (60)  0
\begin{equation}
\label{eq20}0\end{equation}
Type: AlgebraicNumber?

... --aUser,  Thu, 01 Feb 2018 15:35:28 +0000 reply
fricas
f(a) == ( local b; b:=sqrt(a); b^2-a)

   Compiled code for f has been cleared.
   1 old definition(s) deleted for function or rule f
Type: Void
fricas
f(sqrt(2))
fricas
Compiling function f with type AlgebraicNumber -> AlgebraicNumber 

.eq set blank @
:df.
0 
:edf.

      R62=0

   (62)  0
\begin{equation}
\label{eq21}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output fortran on

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

      R63=0

   (63)  0
\begin{equation}
\label{eq22}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output fortran off
 
fricas
)set output tex on 

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

   (64)  0
\begin{equation}
\label{eq23}0\end{equation}
Type: AlgebraicNumber?
fricas
)set output tex off
 
fricas
)set output script on

f(sqrt(2))

.eq set blank @
:df.
0 
:edf.

   (65)  0
Type: AlgebraicNumber?
fricas
)set output script off

test --jr,  Sat, 03 Feb 2018 13:20:20 +0000 reply
((1+2)/3)::SquareMatrix?(3,Integer)
... --jr,  Sat, 03 Feb 2018 13:21:15 +0000 reply
fricas
((1+2)/3)::SquareMatrix(3,Integer)

         +1  0  0+
         |       |
   (66)  |0  1  0|
         |       |
         +0  0  1+
Type: SquareMatrix?(3,Integer)

... --jr,  Sat, 03 Feb 2018 13:21:37 +0000 reply
fricas
((1+2)/3)::SquareMatrix(3,Integer)

         +1  0  0+
         |       |
   (67)  |0  1  0|
         |       |
         +0  0  1+
Type: SquareMatrix?(3,Integer)

integrate(exp(x-x^2), x) --test  int,  Sun, 10 Jun 2018 12:51:40 +0000 reply
fricas
integrate((x+sin(x))^(1/2), x)

   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)

integrate(exp(x-x^2), x) --test  int,  Sun, 10 Jun 2018 12:52:12 +0000 reply\begin{fricas}
integrate((x+sin(x))^(1/2), x)
\end{fricas}

Some or all expressions may not have rendered properly, because Latex returned the following error:

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entering extended mode
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Babel <v3.8h> and hyphenation patterns for english, usenglishmax, dumylang, noh
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ak, danish, dutch, finnish, basque, french, german, ngerman, ibycus, greek, mon
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(/usr/share/texmf-texlive/tex/latex/base/t1enc.def))
(/usr/share/texmf-texlive/tex/latex/pstricks/pstricks.sty
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                           (1/2), x)
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trying something difficult; see comments --h-x, Sat, 09 Nov 2019 02:31:34 +0000 reply

fricas

"";
  "Notice how the CAS computes the second (displayed) integral, while saying 'proved non-elementary' about the first one,";
  "despite the fact that the two are equal for any x in [-1..-1/sqrt(2)].";
  "My guess is that the system does not apply transformations that force additional restrictions on the function domain...";
  "Notice also how transforming the integrand into a form that the CAS can find an elementary antiderivative for requires";
  "quite some discipline in keeping track of the integrand domain, as well as using some not-so-obvious transfromations...";
  t:=sqrt(1-x*x);
  nume:=(acos(x)*t)^(-1);
  deno:=log(2*acos(x)/%pi);
  transformedDeno:=log(2+asin(2*x*t)/%pi);
  failedIntegral:=integrate(nume/transformedDeno,x);
  integrate(nume/deno, x)

$\label{eq1}-{\log \left({\log \left({{2 \ {\arccos \left({x}\right)}}\over \pi}\right)}\right)}$

(1)

Type: Union(Expression(Integer),...)

Still no CAS knows if this one is elementary... Probably it is not, but a fun fact... --h-x, Sat, 09 Nov 2019 02:35:01 +0000 reply

fricas

integrate((x+sin(x))^(1/3),x)

   >> Error detected within library code:
   integrate: implementation incomplete (constant residues)