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

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

fricas
integrate(1/(a+z^3), z=0..1,"noPole")

\label{eq1}{\left(
\begin{array}{@{}l}
\displaystyle
-{{\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)}}+ 
\
\
\displaystyle
{{12}\ {\arctan \left({{{2 \ {\sqrt{3}}\ {\root{3}\of{{a}^{2}}}}-{a \ {\sqrt{3}}}}\over{3 \  a}}\right)}}+{2 \  \pi}
(1)
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

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

...
Wed, 28 Nov 2007 11:15:39 -0800 reply
fricas
integrate(1/sqrt(1 + x^2))
There are 4 exposed and 2 unexposed library operations named integrate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named integrate with argument type(s) Expression(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.
...
Wed, 28 Nov 2007 11:16:29 -0800 reply
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) +------------+ 2 | 2 +------------+ +-+ (32cos(t) - 24)sin(t)\|2cos(t) - 1 | 2 \|2 atan(--------------------------------------) + 8sin(t)\|2cos(t) - 1 +-+ 4 +-+ 2 +-+ 32\|2 cos(t) - 48\|2 cos(t) + 17\|2 ------------------------------------------------------------------------- 16
Type: Union(Expression(Integer),...)
fricas
draw(F1,t=-%pi/4..%pi/4)
fricas
Compiling function %Y 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)
(10) "potentialPole"
Type: Union(pole: potentialPole,...)
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 1.2.3 compiled at Sat May 3 02:17:51 UTC 2014"

fricas
solve(x*b -3*a*b + a*x - 9*b*b-4*b*x = +a*a - 9*a*b ,x)
There are no library operations named + having 1 argument(s) though there are 11 exposed operation(s) and 8 unexposed operation(s) having a different number of arguments. Use HyperDoc Browse, or issue )what op + to learn what operations contain " + " in their names, or issue )display op + to learn more about the available operations.
Cannot find a definition or applicable library operation named + with argument type(s) Polynomial(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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

fricas
solve(-1*(a+3*b)**2 - 3*b*x +a*x = 0,x)
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) Polynomial(Integer) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
solve(-1*(a-3*b)**2 - 3*b*x +a*x = 0,x)
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) Polynomial(Integer) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
solve((a-3*b)(x-a+3*b) = 0= 0,x)
There are 1 exposed and 1 unexposed library operations named elt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op elt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find application of object of type Polynomial(Integer) to argument(s) of type(s) Polynomial(Integer)

fricas
solve((a-3*b)(x-a+3*b) = 0,x)
There are 1 exposed and 1 unexposed library operations named elt having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op elt to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find application of object of type Polynomial(Integer) to argument(s) of type(s) Polynomial(Integer)

fricas
solve((a-3*b)*(x-a+3*b) = 0= 0,x)
There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named solve with argument type(s) Equation(Equation(Polynomial(Integer))) Variable(x)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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

fricas
integrate(1/(a+z^3), z=0..1, "noPole")
(16) +--+2 +--+ +-+ 2 3| 2 3 2 3| 2 4 3 - \|3 log(3a \|a + (- 2a + a )\|a + a - 2a ) + +--+ +--+2 +--+ +-+3| 2 +-+ +-+ 3| 2 3| 2 2 2\|3 \|a - a\|3 2\|3 log(\|a + 2a\|a + a ) + 12atan(------------------) + 2%pi 3a / +--+ +-+3| 2 12\|3 \|a
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

x*y

fricas
a : (INT,INT) := (2,3)
The constructor Tuple takes 1 argument and you have given 2 .

Coherent type inference --gdr, Fri, 02 May 2008 17:11:17 -0700 reply
fricas
(1,2)
(17) [1,2]
Type: Tuple(PositiveInteger)

fricas
('Mon,'Tue)
(18) [Mon,Tue]
Type: Tuple(OrderedVariableList([Mon,Tue]))

fricas
a := 'x :: OutputForm
(19) x
fricas
b := 'y :: OutputForm
(20) y
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
  deq := D(y(x), x, 2) + D(y(x), x) + y(x) + cos(y(x)) = 0
  solve(deq, y, x)
There are no library operations named y Use HyperDoc Browse or issue )what op y to learn if there is any operation containing " y " in its name.
Cannot find a definition or applicable library operation named y with argument type(s) Variable(x)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
y := operator y;
fricas
deq := D(y(x), x, 2) + D(y(x), x) + y(x) + cos(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;
There are 2 exposed and 10 unexposed library operations named operator having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op operator to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named operator with argument type(s) BasicOperator
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. 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)
(5) - 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 (6) ----------------- +-+ +-+ \|b + \|a
Type: Expression(Integer)

Should it be equal to ``1''? --FlowProblem?, Tue, 04 Nov 2008 22:12:49 -0800 reply
fricas
simplify ( sin(a)^2+sin(%pi/2-a)^2 )
%pi - 2a 2 2 (7) - cos(--------) - cos(a) + 2 2
Type: Expression(Integer)

I was mistaken with factor example: I didn't see the ``-'' sign :)

fricas
d:=sin(series(asin(x)))-x
21 (8) O(x )
Type: UnivariatePuiseuxSeries?(Expression(Integer),x,0)
fricas
1.2*d
21 (9) O(x )
Type: UnivariatePuiseuxSeries?(Expression(Float),x,0.0)

This seems to be a bug.

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 (10) [m= %e ]
Type: List(Equation(Expression(Integer)))

quicksort average --grinder, Wed, 26 Nov 2008 13:49:12 -0800 reply
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 (11) [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 (12) [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 (13) -- 3e
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

Bessel Integral

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

fricas
integrate(x^2,x)
1 3 (15) - x 3
Type: Polynomial(Fraction(Integer))

integrate(x/sqrt(x^4+10x^2-96x-71,x)

fricas
integrate(x/sqrt(x^4+10*x^2-96*x-71,x)
Line 1: integrate(x/sqrt(x^4+10*x^2-96*x-71,x) .........A...........................B Error A: Missing mate. Error B: syntax error at top level Error B: Possibly missing a ) 3 error(s) parsing

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

fricas
integrate(a/x^3, x)
a (19) - --- 2 2x
Type: Union(Expression(Integer),...)

9*9

fricas
integrate(1/(a+z^3), z=0..1,"noPole")
(20) +--+2 +--+ +-+ 2 3| 2 3 2 3| 2 4 3 - \|3 log(3a \|a + (- 2a + a )\|a + a - 2a ) + +--+ +--+2 +--+ +-+3| 2 +-+ +-+ 3| 2 3| 2 2 2\|3 \|a - a\|3 2\|3 log(\|a + 2a\|a + a ) + 12atan(------------------) + 2%pi 3a / +--+ +-+3| 2 12\|3 \|a
Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas
integrate(x/(sqrt((x^4)+(10*(x^2))+(96*x)-71)), x)
(21) 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(x/(sqrt((x^4)+(10*(x^2))+(96*x)-71)), x )
(22) 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),...)

solve(a^2+b^2=c^2,c)

fricas
solve(a^2+b^2=c^2,c)
2 2 2 (23) [c - b - a = 0]
Type: List(Equation(Fraction(Polynomial(Integer))))

fricas
integrate(1/sqrt(tan(x)),x)
(24) - +-+ 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),...)

A hard to compute integral --PAP, Tue, 28 Sep 2010 17:55:14 -0700 reply
fricas
a:=simplify(integrate(1/sqrt(tan(x))))
    b:=simplify(differentiate(a),x)
The form on the left hand side of an assignment must be a single variable, a Tuple of variables or a reference to an entry in an object supporting the setelt operation.

A hard to compute integral --PAP, Tue, 28 Sep 2010 17:57:47 -0700 reply
fricas
a:=simplify(integrate(1/sqrt(tan(x))),x);
There are 4 exposed and 2 unexposed library operations named integrate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op integrate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named integrate with argument type(s) Expression(Integer)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. b:=simplify(differentiate(a),x);
There are 3 exposed and 0 unexposed library operations named differentiate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op differentiate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named differentiate with argument type(s) Variable(a)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
a:=simplify(integrate(1/sqrt(tan(x)),x)
Line 1: a:=simplify(integrate(1/sqrt(tan(x)),x) ...........A..........................B Error A: Missing mate. Error B: syntax error at top level Error B: Possibly missing a ) 3 error(s) parsing b:=simplify(differentiate(a),x)
There are 3 exposed and 0 unexposed library operations named differentiate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op differentiate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named differentiate with argument type(s) Variable(a)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
b:=simplify(differentiate(a),x))
Line 1: b:=simplify(differentiate(a),x)) ...............................A Error A: Improper syntax. 1 error(s) parsing




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