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SandBoxFriCAS2
  
last edited 2 years ago by TheDoctor

Try FriCAS calculations here. For example, here is a simple FriCAS command: 

    \begin{axiom}
    integrate(1/(a+z^3), z=0..1,"noPole")
    \end{axiom}

fricas
(1) -> integrate(1/(a+z^3), z=0..1,"noPole")
\label{eq1}\frac{-{{\sqrt{3}}\ {\log \left({{3 \ {{a}^{2}}\ {{\root{3}\of{{a}^{2}}}^{2}}}+{{\left(-{2 \ {{a}^{3}}}+{{a}^{2}}\right)}\ {\root{3}\of{{a}^{2}}}}+{{a}^{4}}-{2 \ {{a}^{3}}}}\right)}}+{2 \ {\sqrt{3}}\ {\log \left({{{\root{3}\of{{a}^{2}}}^{2}}+{2 \ a \ {\root{3}\of{{a}^{2}}}}+{{a}^{2}}}\right)}}+{{12}\ {\arctan \left({\frac{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}{3 \ a}}\right)}}+{{\sqrt{3}}\ {\log \left({{a}^{4}}\right)}}-{2 \ {\sqrt{3}}\ {\log \left({{a}^{2}}\right)}}+{2 \ \pi}}{{12}\ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}(1)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Remember to type \begin{axiom} before each group of commands and \end{axiom} after the commands.

fricas
integrate(1/sqrt(1+x^2), x)
\label{eq2}-{\log \left({{\sqrt{{{x}^{2}}+ 1}}- x}\right)}(2)
Type: Union(Expression(Integer),...)

fricas
)set output algebra on
fricas
)set output tex off

F1:=integrate(cos(t)*sqrt(cos(2*t)),t)

   (3)
                      +-+      4       +-+      2       +-+
          +-+     32 \|2 cos(t)  - 48 \|2 cos(t)  + 17 \|2
       - \|2 atan(-----------------------------------------)
                                           +-------------+
                             2             |        2
                   (32 cos(t)  - 24)sin(t)\|2 cos(t)  - 1
     + 
                +-------------+
                |        2
       8 sin(t)\|2 cos(t)  - 1
  /
     16
Type: Union(Expression(Integer),...)
fricas
draw(F1,t=-%pi/4..%pi/4)
fricas
Compiling function %BE with type DoubleFloat -> DoubleFloat 
   Graph data being transmitted to the viewport manager...
   FriCAS2D data being transmitted to the viewport manager...

   (4)  TwoDimensionalViewport: "FriCAS2D"
Type: TwoDimensionalViewport?
fricas
integrate(cos(t)*sqrt(cos(2*t)),t=-%pi/4..%pi/4)

   (5)  "potentialPole"
Type: Union(pole: potentialPole,...)

fricas
integrate(abs(x),x=0..1)

   (6)  "potentialPole"
Type: Union(pole: potentialPole,...)
fricas
integrate(abs(x),x=0..1,"noPole")

   (7)  "failed"
Type: Union(fail: failed,...)

fricas
integrate(abs(x),x=-1..1)

   (8)  "potentialPole"
Type: Union(pole: potentialPole,...)
fricas
integrate(abs(x),x=-1..1,"noPole")

   (9)  "failed"
Type: Union(fail: failed,...)

fricas
integrate(sqrt(x^2),x=0..1)

         1
   (10)  -
         2
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)
fricas
integrate(sqrt(x^2),x=0..1,"noPole")

         1
   (11)  -
         2
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas
integrate(sqrt(x^2),x=-1..1)

   (12)  "potentialPole"
Type: Union(pole: potentialPole,...)
fricas
integrate(sqrt(x^2),x=-1..1,"noPole")

   (13)  0
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas
)version

"FriCAS 1.3.10 compiled at Wed 10 Jan 02:19:45 CET 2024"

... --schweika, Mon, 31 Dec 2007 07:13:59 -0800 reply
fricas
solve(x*b -3*a*b + a*x - 9*b*b-4*b*x = a*a - 9*a*b ,x)

   (14)  [x = - 3 b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a+3*b)^2 - 3*b*x -a*x = 0,x)

   (15)  [x = - 3 b - a]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a+3*b)^2 - 3*b*x +a*x = 0,x)

                   2            2
              - 9 b  - 6 a b - a
   (16)  [x = -------------------]
                    3 b - a
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a-3*b)^2 - 3*b*x +a*x = 0,x)

   (17)  [x = - 3 b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve((a-3*b)*(x-a+3*b) = 0,x)

   (18)  [x = - 3 b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))

Coherent type inference --gdr, Thu, 21 Feb 2008 20:30:01 -0800 reply
fricas
a : (INT,INT) := (2,3)

   The constructor Tuple takes 1 argument and you have given  2  .

fricas
(1,2)

   (19)  [1, 2]
Type: Tuple(PositiveInteger?)

fricas
('Mon,'Tue)

   (20)  [Mon, Tue]
Type: Tuple(OrderedVariableList([Mon,Tue]))

fricas
a := 'x :: OutputForm

   (21)  x
Type: OutputForm?
fricas
b := 'y :: OutputForm

   (22)  y
Type: OutputForm?
fricas
a and b

   Argument number 1 to "and" must be a Boolean.

fricas
)clear all

   All user variables and function definitions have been cleared.

... --artjcoder, Tue, 13 May 2008 02:27:30 -0700 reply
fricas
y := operator 'y

   (1)  y
Type: BasicOperator?
fricas
deq := D(y(x), x, 2) + D(y(x), x) + y(x) + cos(y(x)) = 0

         ,,       ,
   (2)  y  (x) + y (x) + cos(y(x)) + y(x) = 0
Type: Equation(Expression(Integer))
fricas
solve(deq, y, x)

   >> Error detected within library code:
   parseLODE: not a linear ordinary differential equation

fricas
y := operator 'y;
Type: BasicOperator?
fricas
deq := D(y(x), x, 2) + D(y(x), x) + y(x) + 1 = 0;
Type: Equation(Expression(Integer))
fricas
solve(deq, y, x);
Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...)

exports --gdr, Mon, 26 May 2008 14:22:55 -0700 reply
SandBoxMaybe

Factor --FlowProblem?, Tue, 04 Nov 2008 21:59:44 -0800 reply
fricas
factor(1-x^900)

   (6)
   -
                        2           2       2           4    3    2
        (x - 1)(x + 1)(x  - x + 1)(x  + 1)(x  + x + 1)(x  - x  + x  - x + 1)
     *
          4    2       4    3    2           6    3       6    3
        (x  - x  + 1)(x  + x  + x  + x + 1)(x  - x  + 1)(x  + x  + 1)
     *
          8    7    5    4    3           8    6    4    2
        (x  - x  + x  - x  + x  - x + 1)(x  - x  + x  - x  + 1)
     *
          8    7    5    4    3           12    6
        (x  + x  - x  - x  - x  + x + 1)(x   - x  + 1)
     *
          16    14    10    8    6    2       20    15    10    5
        (x   + x   - x   - x  - x  + x  + 1)(x   - x   + x   - x  + 1)
     *
          20    15    10    5       24    21    15    12    9    3
        (x   + x   + x   + x  + 1)(x   - x   + x   - x   + x  - x  + 1)
     *
          24    21    15    12    9    3
        (x   + x   - x   - x   - x  + x  + 1)
     *
          40    35    25    20    15    5       40    30    20    10
        (x   - x   + x   - x   + x   - x  + 1)(x   - x   + x   - x   + 1)
     *
          40    35    25    20    15    5
        (x   + x   - x   - x   - x   + x  + 1)
     *
          48    42    30    24    18    6
        (x   + x   - x   - x   - x   + x  + 1)
     *
          80    70    50    40    30    10
        (x   + x   - x   - x   - x   + x   + 1)
     *
          120    105    75    60    45    15
        (x    - x    + x   - x   + x   - x   + 1)
     *
          120    105    75    60    45    15
        (x    + x    - x   - x   - x   + x   + 1)
     *
          240    210    150    120    90    30
        (x    + x    - x    - x    - x   + x   + 1)
Type: Factored(Polynomial(Integer))

... --FlowProblem?, Tue, 04 Nov 2008 22:05:49 -0800 reply
fricas
simplify((a+b+2*sqrt(a)*sqrt(b))/(sqrt(a)+sqrt(b)))

           +-+ +-+
        2 \|a \|b  + b + a
   (7)  ------------------
             +-+    +-+
            \|b  + \|a
Type: Expression(Integer)

fricas
simplify ( sin(a)^2+sin(%pi/2-a)^2 )

              %pi - 2 a 2         2
   (8)  - cos(---------)  - cos(a)  + 2
                  2
Type: Expression(Integer)

fricas
d:=sin(series(asin(x)))-x

           21
   (9)  O(x  )
Type: UnivariatePuiseuxSeries?(Expression(Integer),x,0)
fricas
1.2*d

            21
   (10)  O(x  )
Type: UnivariatePuiseuxSeries?(Expression(Float),x,0.0)

quicksort average --grinder, Wed, 26 Nov 2008 13:29:45 -0800 reply
fricas
solve( n*log(n)/log(m) = 1/4*n*log(n/4)/log(m) + 3/4*n*log(3/4*n)/log(m) + 15/4*n + 8, m)

                                     3 n          n
                4 n log(n) - 3 n log(---) - n log(-)
                                      4           4
                ------------------------------------
                              15 n + 32
   (11)  [m = %e                                    ]
Type: List(Equation(Expression(Integer)))

fricas
solve( log(n)/log(m) = 1/4*log(n/4)/log(m) + 3/4*log(3/4*n)/log(m) + 15/4*n + 8, m)

                                 3 n        n
                4 log(n) - 3 log(---) - log(-)
                                  4         4
                ------------------------------
                           15 n + 32
   (12)  [m = %e                              ]
Type: List(Equation(Expression(Integer)))

fricas
solve( log(n)/log(m) = log(n/4)/log(m) + log(3/4*n)/log(m) + 15/4*n + 8, m)

                                 3 n          n
                4 log(n) - 4 log(---) - 4 log(-)
                                  4           4
                --------------------------------
                            15 n + 32
   (13)  [m = %e                                ]
Type: List(Equation(Expression(Integer)))

simple integration --kylekanos, Tue, 09 Dec 2008 09:55:35 -0800 reply
fricas
integrate((P*cos(x))/(2*e)*cos(x)*sin(x), x=0..%pi,"noPole")

          P
   (14)  ---
         3 e
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Bessel Integral

fricas
integrate(exp(z*cos(t)),t=0..2*%pi)

   (15)  "failed"
Type: Union(fail: failed,...)

... --anonymous, Wed, 08 Apr 2009 23:10:17 -0700 reply
fricas
integrate(x/sqrt(x^4+10*x^2-96*x-71),x)

   (16)
     log
                                                     +----------------------+
            6       4       3       2                | 4       2
          (x  + 15 x  - 80 x  + 27 x  - 528 x + 781)\|x  + 10 x  - 96 x - 71
        + 
           8       6        5       4         3         2
          x  + 20 x  - 128 x  + 54 x  - 1408 x  + 3124 x  + 10001
  /
     8
Type: Union(Expression(Integer),...)

... --Ric, Fri, 10 Apr 2009 14:05:50 -0700 reply
fricas
integrate(1/x, x=1..2)

         log(4)
   (17)  ------
            2
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

beginner --Ric, Fri, 10 Apr 2009 14:08:30 -0700 reply
fricas
integrate(a/x^3, x)

             a
   (18)  - ----
              2
           2 x
Type: Union(Expression(Integer),...)

sixth --Tom, Tue, 02 Jun 2009 19:29:03 -0700 reply
fricas
solve(a^2+b^2=c^2,c)

           2    2    2
   (19)  [c  - b  - a  = 0]
Type: List(Equation(Fraction(Polynomial(Integer))))

A hard to compute integral --PAP, Tue, 28 Sep 2010 17:55:14 -0700 reply
fricas
integrate(1/sqrt(tan(x)), x)

   (20)
        +-+     +-+ +------+
       \|2 log(\|2 \|tan(x)  + tan(x) + 1)
     + 
                                                               +------+    +-+
          +-+       +-+ +------+                    +-+     2 \|tan(x)  + \|2
       - \|2 log(- \|2 \|tan(x)  + tan(x) + 1) + 2 \|2 atan(------------------)
                                                                    +-+
                                                                   \|2
     + 
                     +------+    +-+
          +-+     2 \|tan(x)  - \|2
       2 \|2 atan(------------------)
                          +-+
                         \|2
  /
     4
Type: Union(Expression(Integer),...)
fricas
a :=simplify(integrate(1/sqrt(tan(x)), x))

   (21)
                          +------+
                +-+       |sin(x)
               \|2 cos(x) |------  + sin(x) + cos(x)
        +-+              \|cos(x)
       \|2 log(-------------------------------------)
                               cos(x)
     + 
                              +------+
                    +-+       |sin(x)
                 - \|2 cos(x) |------  + sin(x) + cos(x)
          +-+                \|cos(x)
       - \|2 log(---------------------------------------)
                                  cos(x)
     + 
                     +------+                         +------+
                     |sin(x)     +-+                  |sin(x)     +-+
                  2  |------  + \|2                2  |------  - \|2
          +-+       \|cos(x)               +-+       \|cos(x)
       2 \|2 atan(------------------) + 2 \|2 atan(------------------)
                          +-+                              +-+
                         \|2                              \|2
  /
     4
Type: Expression(Integer)
fricas
b:=simplify(differentiate(a, x))

                +------+
                |sin(x)
         cos(x) |------
               \|cos(x)
   (22)  ---------------
              sin(x)
Type: Expression(Integer)
fricas
normalize(b - 1/sqrt(tan(x)))

   (23)  0
Type: Expression(Integer)



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