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SandBoxFriCAS

last edited 1 year ago by test1

Edit detail for SandBoxFriCAS revision 57 of 62

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Editor: minh Time: 2022/04/20 17:01:32 GMT+0
Note: Test

added:

From minh Wed Apr 20 17:01:31 +0000 2022
From: minh
Date: Wed, 20 Apr 2022 17:01:31 +0000
Subject: Test
Message-ID: <20220420170131+0000@fricas-wiki.math.uni.wroc.pl>

\begin{equation}
guess([3,6,9,12,15,18,21,24,27,30])
\end{equation}

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

Value = "FriCAS 1.3.7 compiled at Wed Jun 30 16:44:06 UTC 2021"

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{{{v}^{2}}+{2 \  a \  s}}}- v}\over a}}, \:{t ={{{\sqrt{{{v}^{2}}+{2 \  a \  s}}}- v}\over 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.

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 may need to 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 may need to 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 scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

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

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value on is not among the valid choices.

f(sqrt(2))

      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 

      R52=0

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

f(sqrt(2))

      R53=0

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

f(sqrt(2))

      R54=0

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

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value console is not among the valid choices.

f(sqrt(2))

      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 

      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))

      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))

      R59=0

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

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value on is not among the valid choices.

fricas
)set output script console

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value console is not among the valid choices.

f(sqrt(2))

      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 

      R62=0

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

f(sqrt(2))

      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))

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

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value on is not among the valid choices.

f(sqrt(2))

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

--------------------------- The scripts Option ----------------------------

 Description: show subscripts,... linearly

 The scripts option may be followed by any one of the following:

    yes
    no

 The current setting is indicated within the list.

 Your value off is not among the valid choices.

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 (has polynomial part)

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}
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)

                   2 acos(x)
   (68)  - log(log(---------))
                      %pi
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 (has polynomial part)

Axiom is lying here, the example comes from Bronstein's tutorial,
        pp. 29-31 --h-x,  Sun, 10 Nov 2019 09:26:19 +0000 reply
fricas
t:=x+log(x);
 nume:=(x^2+x+1)*sqrt(t)+(3*x+1)*log(x)+3*x*x+x;
 deno:=x*t*sqrt(t)+x*x*t;
 check:=integrate(nume/deno,x);
 differentiate(2*sqrt(t)+2*log(x+sqrt(t)), x)

           2            +----------+                        2
         (x  + 2 x + 1)\|log(x) + x  + (3 x + 1)log(x) + 3 x  + x
   (69)  --------------------------------------------------------
                            2  +----------+    2          3
               (x log(x) + x )\|log(x) + x  + x log(x) + x
Type: Expression(Integer)

No lie, compare:
fricas
nume - numer(differentiate(2*sqrt(t)+2*log(x+sqrt(t)), x))

             +----------+
   (70)  - x\|log(x) + x
Type: Expression(Integer)

Still, shame that FriCAS? can not do corrected example...
Axiom fails again,
        see Bronstein's Tutorial pp. 32-33 for details on this example --h-x,  Sun, 10 Nov 2019 13:21:41 +0000 reply
fricas
nume:= 3*(x+exp(x))^(1/3)+(2*x*x+3*x)*exp(x)+5*x*x;
 deno:= x*(x+exp(x))^(1/3);
 "integrate(nume/deno, x)"
 (q:=nume/deno,  differentiate(3*x*(x+exp(x))^(2/3)+3*log(x), x))

   (71)
       +-------+
      3|  x            2         x      2
    3 \|%e  + x  + (2 x  + 3 x)%e  + 5 x
   [-------------------------------------,
                   +-------+
                  3|  x
                 x\|%e  + x
       +-------+
      3|  x            2         x      2
    3 \|%e  + x  + (2 x  + 3 x)%e  + 5 x
    -------------------------------------]
                   +-------+
                  3|  x
                 x\|%e  + x
Type: Tuple(Expression(Integer))

testing --h-x,  Sun, 02 Feb 2020 04:20:03 +0000 reply
fricas
)clear value t

integrate(sqrt(A-2*t*t/((1-t*t)^2))*2/(1+t*t), t)

   (72)
   [
            +---------+
           \|- 4 A - 2
        *
           log
                                           +------------------------+
                                           |   4               2
                      3        +---------+ |A t  + (- 2 A - 2)t  + A       4
                  (2 t  - 2 t)\|- 4 A - 2  |------------------------  + A t
                                           |       4      2
                                          \|      t  - 2 t  + 1
                + 
                              2
                  (- 6 A - 4)t  + A
             /
                 4      2
                t  + 2 t  + 1
       + 
                               +------------------------+
                               |   4               2
                        2      |A t  + (- 2 A - 2)t  + A
                      (t  - 1) |------------------------
                               |       4      2
              +-+             \|      t  - 2 t  + 1
         - 2 \|2 atan(-----------------------------------)
                                       +-+
                                     t\|2
    /
       4
     ,

                                        +-------+
            +-------+                 t\|4 A + 2
         - \|4 A + 2 atan(-----------------------------------)
                                   +------------------------+
                                   |   4               2
                            2      |A t  + (- 2 A - 2)t  + A
                          (t  - 1) |------------------------
                                   |       4      2
                                  \|      t  - 2 t  + 1
       + 
                             +------------------------+
                             |   4               2
                      2      |A t  + (- 2 A - 2)t  + A
                    (t  - 1) |------------------------
                             |       4      2
            +-+             \|      t  - 2 t  + 1
         - \|2 atan(-----------------------------------)
                                     +-+
                                   t\|2
    /
       2
     ]
Type: Union(List(Expression(Integer)),...)

... --TheDoctor?,  Mon, 31 Jan 2022 23:17:14 +0000 reply
fricas
integrate(x/sqrt(2*%pi)*exp(-1/2*log(x)^2),x=0..%plusInfinity)

           2 +-+ +---+
         %e \|2 \|%pi
   (73)  -------------
             +-----+
            \|2 %pi
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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                $
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                                       2-a)
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                $
l.184 integrate((x+sin(x))^
                           (1/2), x)
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[25] (./626913819321871499-16.0px.aux) )
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Transcript written on 626913819321871499-16.0px.log.

Test --minh, Wed, 20 Apr 2022 16:58:21 +0000 reply

guess([3,6,9,12,15,18,21,24,27,30]?)

Test --minh,  Wed, 20 Apr 2022 17:00:22 +0000 reply\begin{fricas}
guess([3,6,9,12,15,18,21,24,27,30])
\end{fricas}

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... --minh, Wed, 20 Apr 2022 17:01:05 +0000 reply

$\label{eq1} guess([3,6,9,12,15,18,21,24,27,30])$

(1)

Test --minh, Wed, 20 Apr 2022 17:01:31 +0000 reply

$\label{eq1} guess([3,6,9,12,15,18,21,24,27,30])$

(1)