Things we will want to simplify:

2.y.3 -> 6.y
y.x.3 + x^2 +1 -> x^2 + 3*x*y + 1
Zeroes and ones
x + 0 -> x
1*x -> x
x^1 -> x
x^0 -> 1
Simplify fractions
(x^2+5.x+6)/(x+2) -> x+3
Polynomials
(x+1)^2-x^2 -> 2*x+1
Fractions
(2*x)/(x^2-1) - 1/(x-1) -> 1/(x+1)
Functional identities
ln(2x) - ln(x) -> ln(2)
y*sin^2(x) + y*cos^2(x) -> y

Note : The simplification process can be interrupted! -> It probably browses a graph