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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
integrate(1/(a+z^3), z=0..1,"noPole")
 (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)
 (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
- \|2 atan(--------------------------------------) + 8sin(t)\|2cos(t)  - 1
+------------+
2             |       2
(32cos(t)  - 24)sin(t)\|2cos(t)  - 1
---------------------------------------------------------------------------
16
Type: Union(Expression(Integer),...)
fricas
draw(F1,t=-%pi/4..%pi/4)
fricas
Compiling function %BD 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
Value = "FriCAS 1.3.1 compiled at Thu Feb 16 03:24:08 UTC 2017"

fricas
solve(x*b -3*a*b + a*x - 9*b*b-4*b*x = a*a - 9*a*b ,x)
(14)  [x = - 3b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a+3*b)^2 - 3*b*x -a*x = 0,x)
(15)  [x = - 3b - a]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a+3*b)^2 - 3*b*x +a*x = 0,x)
2           2
- 9b  - 6a b - a
(16)  [x = -----------------]
3b - a
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
solve(-1*(a-3*b)^2 - 3*b*x +a*x = 0,x)
(17)  [x = - 3b + a]
Type: List(Equation(Fraction(Polynomial(Integer))))

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

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))),...)

SandBoxMaybe

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))

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 - 2a 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)
3n          n
4n log(n) - 3n log(--) - n log(-)
4          4
---------------------------------
15n + 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)
3n        n
4log(n) - 3log(--) - log(-)
4        4
---------------------------
15n + 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)
3n         n
4log(n) - 4log(--) - 4log(-)
4         4
----------------------------
15n + 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)  --
3e
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Bessel Integral

fricas
integrate(exp(z*cos(t)),t=0..2*%pi)
(15)  "failed"
Type: Union(fail: failed,...)

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

fricas
integrate(1/x, x=1..2)
(17)  log(2)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas
integrate(a/x^3, x)
a
(18)  - ---
2
2x
Type: Union(Expression(Integer),...)

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)
-
+-+
4\|2
*
atan
1
-----------------------------------------------------------------
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
|             \|cos(x)                          +-+ |sin(x)
|------------------------------------------  + \|2  |------  - 1
\|                  cos(x)                          \|cos(x)
+
-
+-+
4\|2
*
1
atan(---------------------------------------------------------------)
+----------------------------------------+
|            +------+
|  +-+       |sin(x)
|2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
|           \|cos(x)                          +-+ |sin(x)
|----------------------------------------  + \|2  |------  + 1
\|                 cos(x)                         \|cos(x)
+
+------+
+-+       |sin(x)
2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
+-+               \|cos(x)
\|2 log(----------------------------------------)
cos(x)
+
+------+
+-+       |sin(x)
- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
+-+                 \|cos(x)
- \|2 log(------------------------------------------)
cos(x)
/
4
Type: Union(Expression(Integer),...)
fricas
a :=simplify(integrate(1/sqrt(tan(x)), x))
(21)
-
+-+
4\|2
*
atan
1
-----------------------------------------------------------------
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
|             \|cos(x)                          +-+ |sin(x)
|------------------------------------------  + \|2  |------  - 1
\|                  cos(x)                          \|cos(x)
+
-
+-+
4\|2
*
1
atan(---------------------------------------------------------------)
+----------------------------------------+
|            +------+
|  +-+       |sin(x)
|2\|2 cos(x) |------  + 2sin(x) + 2cos(x)         +------+
|           \|cos(x)                          +-+ |sin(x)
|----------------------------------------  + \|2  |------  + 1
\|                 cos(x)                         \|cos(x)
+
+------+
+-+       |sin(x)
2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
+-+               \|cos(x)
\|2 log(----------------------------------------)
cos(x)
+
+------+
+-+       |sin(x)
- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
+-+                 \|cos(x)
- \|2 log(------------------------------------------)
cos(x)
/
4
Type: Expression(Integer)
fricas
b:=simplify(differentiate(a, x))
(22)
+------+
3                 4           2  |sin(x)
(48cos(x) sin(x) + 39cos(x)  - 64cos(x) ) |------
\|cos(x)
*
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|             \|cos(x)
|------------------------------------------
\|                  cos(x)
+
+------+
3                 4           2  |sin(x)
(- 64cos(x) sin(x) - 29cos(x)  + 64cos(x) ) |------
\|cos(x)
+
+-+      3      +-+                   +-+      4      +-+      2
(45\|2 cos(x)  - 64\|2 cos(x))sin(x) - 32\|2 cos(x)  + 32\|2 cos(x)
*
+----------------------------------------+
|            +------+
|  +-+       |sin(x)
|2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|           \|cos(x)
|----------------------------------------
\|                 cos(x)
+
+------+
3                 4           2  |sin(x)
(64cos(x) sin(x) + 29cos(x)  - 64cos(x) ) |------
\|cos(x)
+
+-+      3      +-+                   +-+      4      +-+      2
(45\|2 cos(x)  - 64\|2 cos(x))sin(x) - 32\|2 cos(x)  + 32\|2 cos(x)
*
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|             \|cos(x)
|------------------------------------------
\|                  cos(x)
+
+------+
3                              4           2  |sin(x)
((14cos(x)  - 128cos(x))sin(x) - 47cos(x)  + 96cos(x) ) |------
\|cos(x)
/
3                             4           2
((39cos(x)  - 64cos(x))sin(x) - 48cos(x)  + 48cos(x) )
*
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|             \|cos(x)
|------------------------------------------
\|                  cos(x)
+
+-+      3      +-+                   +-+      4
(45\|2 cos(x)  - 64\|2 cos(x))sin(x) - 32\|2 cos(x)
+
+-+      2
32\|2 cos(x)
*
+------+
|sin(x)
|------
\|cos(x)
+
3                             4           2
(- 29cos(x)  + 64cos(x))sin(x) + 64cos(x)  - 64cos(x)
*
+----------------------------------------+
|            +------+
|  +-+       |sin(x)
|2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|           \|cos(x)
|----------------------------------------
\|                 cos(x)
+
+-+      3      +-+                   +-+      4
(45\|2 cos(x)  - 64\|2 cos(x))sin(x) - 32\|2 cos(x)
+
+-+      2
32\|2 cos(x)
*
+------+
|sin(x)
|------
\|cos(x)
+
3                             4           2
(29cos(x)  - 64cos(x))sin(x) - 64cos(x)  + 64cos(x)
*
+------------------------------------------+
|              +------+
|    +-+       |sin(x)
|- 2\|2 cos(x) |------  + 2sin(x) + 2cos(x)
|             \|cos(x)
|------------------------------------------
\|                  cos(x)
+
3                             4            2
(- 47cos(x)  + 96cos(x))sin(x) - 14cos(x)  + 142cos(x)  - 128
Type: Expression(Integer)
fricas
normalize(b - 1/sqrt(tan(x)))
(23)  0
Type: Expression(Integer)

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