You have to use \begin{axiom} ... \end{axiom} or \begin{reduce} ... \end{reduce} before and after the command like this:

    \begin{reduce}
    int(1/(a+z^3), z);
    \end{reduce}

Oh yes, note that for Axiom you don't end the command with ; and the command for integration in Axiom is integrate.

integrate(1/(a+z^3), z)

(1)

But it must be there for Reduce.

r^2+r+1;

This won't work:

    \begin{axiom}integrate((x^2+2*x*ln(x)+5)/(sin(x^2+x^3-x^4)^2), x)\end{axiom}

Put the \begin{axiom} and \end{axiom} on separate lines and notice that in Axiom ln is written log

integrate((x^2+2*x*log(x)+5)/(sin(x^2+x^3-x^4)^2), x)

(2)

This is wrong:

    \begin{axiom}
    \sqrt{49/100}
    \end{axiom}

Begin each comment with an explanation. Don't put \ in front of the Axiom command.

Do it like this:

    Some explanation
    \begin{axiom}
    sqrt{49/100}
    \end{axiom}

Some explanation

sqrt{49/100}

(3)

This is wrong:

    \begin{axiom}
    $ \sqrt{49/100} $
    \end{axiom}

Don't put $ before and after $ and there is no \ in front.

Just do it like this:

    \begin{axiom}
    sqrt{49/100}
    \end{axiom}

and what you will see is this:

sqrt{49/100}

(4)

This command appears to work

integrate(x^5 ln[x],x)

(5)

But notice that

5 ln[x]

(6)

is something strange. Oddly perhaps, Axiom interprets 5 as a UnivariatePolynomial and 'ln[x]' as a subscripted Symbol and the result is a univariate polynomial in the variable 'ln[x]'.

So perhaps what you meant to write was:

integrate(x^5*log(x),x)

(7)

The command:

    \begin(axiom)
    integrate(sin(x))
    \end(axiom)

wont work.

Use "braces" like this { } not parenthesis ( ) around {axiom}.

Finally, unless the expression is a univariate polynomial, then you must also specify the variable with which to integrate.

integrate(sin(x),x)

(8)

This command:

    \begin{axiom}
    solve{xy=1,x}
    \end{axiom}

uses {} after the solve operation. This is syntactically correct but it probably doesn't do what you might expect.

solve{xy=1,x}

(9)

In Axiom {...,...} is executed as a block of commands which returns the result of the last command in the sequence. Compare

a:={xy=1,x}

(10)

which is just x to

b:=(xy=1,x)

(11)

solve normally operates on such a tuple and

c:=[xy=1,x]

(12)

which is a list and finally

c:=set [xy=1,x]

(13)

which is how to construct a set.

Also notice that multiplication must be written using *

solve(x*y=1,x)

(14)

I'd like to see if Axiom can do my favorite definite integral:

    \begin{axiom}
    integrate(x^4/(sinh(x))^2,x,-inf,inf)
    \end{axiom}

In Axiom use %minusInfinity and %plusInfinity instead of -inf and inf.

integrate(x^4/(sinh(x))^2,x=%minusInfinity..%plusInfinity)

(15)

The results of calculations depend on the type of the inputs You can tell Axiom that you would like the result expressed as a floating point number (if possible) using @. For example:

asin(1/2)@Float

(16)

The trig functions are expressed in radians so use instead and instead of . Finally, because Axiom prefers symbolic calculations express as a rational number

r:Fraction Integer:=1.544

(17)

eq1:=90*%pi/180-asin(n*sin(34*%pi/180)/r)=asin(n/r)

(18)

s:=solve(eq1,n)

(19)

Axiom thinks there are two solutions, unfortunately only one is valid:

eval(eq1,s.1)::Equation Expression Float

(20)

eval(eq1,s.2)::Equation Expression Float

(21)

For example

integrate((4 - x**2)**.5::Expression Fraction Integer, x)

(22)

Since sin(x) cannot be interpreted as a univariate polynomial, you must specify the integration variable.

differentiate(sin(x),x)

(23)