Guiding Mathematica to perform specific simplifications

Question

I would like Mathematica to cancel out a specific term as much as possible. One of my computations simplify to

((Pi*((l^2 + m^2 + 1)
      * ((l^2 + m^2 + 1) * Derivative[0, 1][w][l, m]^2 
         + Derivative[1, 0][w][l, m] * ((l^2 + m^2 + 1) * Derivative[1, 0][w][l, m] - 4*m)
         + 4*l*Derivative[0, 1][w][l, m])
      + 4*(l^2 + m^2)))
  / (l^2 + m^2 + 1))

I would like it to eliminate the l^2 + m^2 + 1 term as much as possible. Distrubute goes part of the way:

 (4 * (l^2 + m^2) * Pi / (1 + l^2 + m^2)^2 
  + ((Pi*(4*l*Derivative[0, 1][w][l, m]
          + (1 + l^2 + m^2) * Derivative[0, 1][w][l, m]^2
          + Derivative[1, 0][w][l, m]
            *(-4*m + (1 + l^2 + m^2)*Derivative[1, 0][w][l, m])))
     / (1 + l^2 + m^2)))

Now I would like to cancel out the two occurences in the numerateor of the second term. I have tried various combinations of Cancel, Apart, Expand, Factor and Simplify, but none of those went in the right direction.

Answer 1

Here is a way to do it, starting with the last expression in your question:

Collect[
   (4*(l^2 + m^2)*
    Pi/(1 + l^2 + 
        m^2)^2 + ((Pi*(4*l*
          Derivative[0, 1][w][l, m] + (1 + l^2 + m^2)*
          Derivative[0, 1][w][l, m]^2 + 
         Derivative[1, 0][w][l, 
           m]*(-4*m + (1 + l^2 + m^2)*
             Derivative[1, 0][w][l, m])))/(1 + l^2 + m^2))), 
 1 + l^2 + m^2]

$$\frac{4 \pi l w^{(0,1)}(l,m)-4 \pi m w^{(1,0)}(l,m)}{l^2+m^2+1}+\pi w^{(0,1)}(l,m)^2+\pi w^{(1,0)}(l,m)^2+\frac{4 \pi \left(l^2+m^2\right)}{\left(l^2+m^2+1\right)^2}$$