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

axiom
ex1:=integrate(log(x)+x, x)
\begin{equation}
\label{eq1}{{2 \  x \ {\log \left({x}\right)}}+{x^2}-{2 \  x}}\over 2\end{equation}
Type: Union(Expression(Integer),...)
axiom
%::InputForm
\begin{equation}
\label{eq2}\left(/ \ {\left(+ \ {\left( \ {\left( \  2 \  x \right)}\ {\left(\log \  x \right)}\right)}\ {\left(+ \ {\left(^\  x \  2 \right)}\ {\left( \  - 2 \  x \right)}\right)}\right)}\  2 \right)\end{equation*}
Type: InputForm
axiom
ex2:=interpret((%::InputForm).2.2)
\begin{equation}
\label{eq3}2 \  x \ {\log \left({x}\right)}\end{equation}
Type: Expression(Integer)

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))
Type: Void

Then manipulating expressions looks like this:
axiom
op(1,ex1)
axiom
Compiling function op with type (PositiveInteger,Expression(Integer)
      ) -> Any
\begin{equation}
\label{eq4}{2 \  x \ {\log \left({x}\right)}}+{x^2}-{2 \  x}\end{equation}
Type: Expression(Integer)
axiom
op(1,%)
axiom
Compiling function op with type (PositiveInteger,Any) -> Any
\begin{equation}
\label{eq5}2 \  x \ {\log \left({x}\right)}\end{equation}
Type: Expression(Integer)
axiom
(op(1,op(1,ex1))-op(2,op(1,ex1)))/op(2,ex1)
\begin{equation}
\label{eq6}{{2 \  x \ {\log \left({x}\right)}}-{x^2}+{2 \  x}}\over 2\end{equation}
Type: Expression(Integer)

Rules and Pattern Matching (from WesterProblemSet)
  Trigonometric manipulations---these are typically difficult for students
axiom
r:= cos(3*x)/cos(x)
\begin{equation}
\label{eq7}{\cos \left({3 \  x}\right)}\over{\cos \left({x}\right)}\end{equation}
Type: Expression(Integer)

 => cos(x)^2 - 3 sin(x)^2 or similar
axiom
real(complexNormalize(r))
\begin{equation}
\label{eq8}-{2 \ {{\sin \left({x}\right)}^2}}+{2 \ {{\cos \left({x}\right)}^2}}- 1\end{equation}
Type: Expression(Integer)

=> 2 cos(2 x) - 1
axiom
real(normalize(simplify(complexNormalize(r))))
\begin{equation}
\label{eq9}{2 \ {\cos \left({2 \  x}\right)}}- 1\end{equation}
Type: Expression(Integer)

Use rewrite rules => cos(x)^2 - 3 sin(x)^2
axiom
sincosAngles:= rule
   cos((n | integer?(n)) * x) ==
      cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x)
   sin((n | integer?(n)) * x) ==
      sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x)
\begin{equation*}
\label{eq10}\begin{array}{@{}l}
\displaystyle
\left\{{{\cos \left({n \  x}\right)}\mbox{\rm = =}{-{{\sin \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\  x}\right)}}+{{\cos \left({x}\right)}\ {\cos \left({{\left(n - 1 \right)}\  x}\right)}}}}, \right.
\
\
\displaystyle
\left.\:{{\sin \left({n \  x}\right)}\mbox{\rm = =}{{{\cos \left({x}\right)}\ {\sin \left({{\left(n - 1 \right)}\  x}\right)}}+{{\cos \left({{\left(n - 1 \right)}\  x}\right)}\ {\sin \left({x}\right)}}}}\right\} \end{array}
\end{equation*}
Type: Ruleset(Integer,Integer,Expression(Integer))

axiom
sincosAngles r
\begin{equation}
\label{eq11}-{3 \ {{\sin \left({x}\right)}^2}}+{{\cos \left({x}\right)}^2}\end{equation}
Type: Expression(Integer)

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)
\begin{equation*}
\label{eq12}\left[{coef = 3}, \:{var = x}\right]?\end{equation*}
Type: Union(Record(coef: Integer,var: Kernel(Expression(Integer))),...)

but
axiom
isMult(x*y)
\begin{equation}
\label{eq13}\mbox{\tt "failed"}\end{equation}
Type: Union("failed",...)

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)
\begin{equation*}
\label{eq14}\left[{val ={x^y}}, \:{exponent = 1}\right]?\end{equation*}
Type: Union(Record(val: Expression(Integer),exponent: Integer),...)

whereas
axiom
isPower(x**10)
\begin{equation*}
\label{eq15}\left[{val = x}, \:{exponent ={10}}\right]?\end{equation*}
Type: Union(Record(val: Expression(Integer),exponent: Integer),...)

In the first case the Integer is assume to be 1.
We have:
axiom
isTimes(x*y*z)
\begin{equation*}
\label{eq16}\left[ z , \: y , \: x \right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)
axiom
isPlus(x+y+z*y)
\begin{equation*}
\label{eq17}\left[{y \  z}, \: y , \: x \right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)

Whereas 
axiom
isTimes((x+y)*z)
\begin{equation}
\label{eq18}\mbox{\tt "failed"}\end{equation}
Type: Union("failed",...)

That is because the expression is internally treated as a
MultivariatePolynomial like this:
axiom
((x+y)*z)::MPOLY([x,y,z],INT)
\begin{equation}
\label{eq19}{z \  x}+{z \  y}\end{equation}
Type: MultivariatePolynomial([x,y,z],Integer)

If you say:
axiom
isPlus((x+y)*z)
\begin{equation*}
\label{eq20}\left[{y \  z}, \:{x \  z}\right]?\end{equation*}
Type: Union(List(Polynomial(Integer)),...)

perhaps the result makes sense?
For some of the details of these operations I consulted the
actual algebra code at:
http://axiom-wiki.newsynthesis.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.

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