The domain InputForm can be quite useful for manipulating parts of expressions. For example

axiom
ex1:=integrate(log(x)+x, x)

(1)

axiom
%::InputForm

(2)

axiom
ex2:=interpret((%::InputForm).2.2)

(3)

If you would like to do this with a more common type of expression and hide the details, you can define

axiom
op(n,x) == interpret((x::InputForm).(n+1))

Then manipulating expressions looks like this:

axiom
op(1,op(1,ex1))

Compiling function op with type (PositiveInteger,Expression Integer)
       -> Any

Compiling function op with type (PositiveInteger,Any) -> Any

(4)

axiom
(op(1,op(1,ex1))-op(2,op(1,ex1)))/op(2,ex1)

(5)

Other Operations

The domain FunctionSpace? includes the following operations:

    isExpt(p,f:Symbol) returns [x, n] if p = x**n and n <> 0 and x = f(a)
    isExpt(p,op:BasicOperator) returns [x, n] if p = x**n and n <> 0 and x = op(a)
    isExpt(p) returns [x, n] if p = x**n and n <> 0
    isMult(p) returns [n, x] if p = n * x and n <> 0
    isPlus(p) returns [m1,...,mn] if p = m1 +...+ mn and n > 1
    isPower(p) returns [x, n] if p = x**n and n <> 0
    isTimes(p) returns [a1,...,an] if p = a1*...*an and n > 1

If these conditions are not met, then the above operations return "failed".

For example,

axiom
isMult(3*x)

(6)

but

axiom
isMult(x*y)

(7)

In the context of Expression Integer, or Polynomial Integer the parameter n must be an Integer. The Symbol y is not an Integer.

Not exactly analogously

axiom
isPower(x**y)

(8)

whereas

axiom
isPower(x**10)

(9)

In the first case the Integer is assume to be 1.

We have:

axiom
isTimes(x*y*z)

(10)

axiom
isPlus(x+y+z*y)

(11)

Whereas

axiom
isTimes((x+y)*z)

(12)

That is because the expression is internally treated as a MultivariatePolynomial like this:

axiom
((x+y)*z)::MPOLY([x,y,z],INT)

(13)

If you say:

axiom
isPlus((x+y)*z)

(14)

perhaps the result makes sense?

For some of the details of these operations I consulted the actual algebra code at:

http://wiki.axiom-developer.org/axiom--test--1/src/algebra/FspaceSpad

Click on pdf or dvi to see the documentation.

You can also enter expressions like isTimes in the search box on the upper right and see all the places in the algebra where this operation is defined and used.