One could consider a home cooked simplify. The following could be packed into a function.
- Step 1
Find the variables:
var = (expr // Variables);
- Step 2
Find which variables occur the most in the multiplications:
{Count[expr, #*_], #} & /@ var // Sort
{{2, G2}, {2, G3}, {2, G6}, {2, G7}, {3, G1}, {4, G4}, {4, G5}, {4, G8}, {5, P2}, {6, P3}, {6, P5}, {6, P6}}
(included transposition to reduce displayed height )
$$\left( \begin{array}{cccccccccccc} 2 & 2 & 2 & 2 & 3 & 4 & 4 & 4 & 5 & 6 & 6 & 6 \\ \text{G2} & \text{G3} & \text{G6} & \text{G7} & \text{G1} & \text{G4} & \text{G5} & \text{G8} & \text{P2} & \text{P3} & \text{P5} & \text{P6} \\ \end{array} \right)$$
- Step 3
Collect the variables that occur the most in multiplications:
simplified = Collect[expr, {P6, P5, P3}, Simplify]
(G3 + G4 + G5 + G7 + G8) P2 + (G1 + G2 + G4 + G5 + G6 + G8) P3 + (G1 + G3 + G4 + G5 + G7 + G8) P5 + (G1 + G2 + G4 + G5 + G6 + G8) P6
$$ \text{P3} (\text{G1}+\text{G2}+\text{G4}+\text{G5}+\text{G6}+\text{G8})+\text{P6} (\text{G1}+\text{G2}+\text{G4}+\text{G5}+\text{G6}+\text{G8})+\text{P5} (\text{G1}+\text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8})+\text{P2} (\text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8}) $$
- Step 4
Rename sub expressions that occur more than once, simplify and revert back to their original expression.
One could maybe use
Experimental`OptimizeExpression
(see for example Common subexpression from two expressions )
to find common sub expressions in the expression above but instead I consider a more simple approach for this kind of structure:
simplified // Level[#, {2}] & // Tally
{{G3 + G4 + G5 + G7 + G8, 1}, {P2, 1}, {G1 + G2 + G4 + G5 + G6 + G8, 2}, {P3, 1}, {G1 + G3 + G4 + G5 + G7 + G8, 1}, {P5, 1}, {P6, 1}}
$$\left( \begin{array}{cc} \text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8} & 1 \\ \text{P2} & 1 \\ \text{G1}+\text{G2}+\text{G4}+\text{G5}+\text{G6}+\text{G8} & 2 \\ \text{P3} & 1 \\ \text{G1}+\text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8} & 1 \\ \text{P5} & 1 \\ \text{P6} & 1 \\ \end{array} \right)$$
G1 + G2 + G4 + G5 + G6 + G8 appears twice we can rename it then simplify then rename again:
Collect[simplified /. G1 + G2 + G4 + G5 + G6 + G8 -> h, h] /.
h -> G1 + G2 + G4 + G5 + G6 + G8
$$(\text{P3}+\text{P6}) (\text{G1}+\text{G2}+\text{G4}+\text{G5}+\text{G6}+\text{G8})+\text{P5} (\text{G1}+\text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8})+\text{P2} (\text{G3}+\text{G4}+\text{G5}+\text{G7}+\text{G8})$$