Finding sub-expressions that simplify larger formula

Note, that if you just propose Mathematica additional equalities, it will give them try:

f = (1/(a^2 (b + m))) (-(-1 + a + 2 b) (-1 + 2 b + 2 m) - 
    4 b ArcTanh[1 - 2 b] + Log[4] + 
    2 ((2 + a) b + m + a m) Log[1 + a] - a (b + m) Log[4 - 4 b] + 
    a Log[(2 - 2 b)/(1 + a)] - m Log[1 - b] + 
    2 Log[(1 - b)/(1 + a)] + (m - a (b + m)) Log[b]);



In[6]:= 
LeafCount[f]
fs = FullSimplify[f, {c == 1 + a}];
LeafCount[fs]


Out[6]= 114

Out[8]= 90

Sometimes you can make a good guess from the additional information you have about you problem (in physics you often do).

One way to find repeating sub expressions:

SortBy[Select[
  Tally[Level[f, -1]],
  LeafCount[First[#]] > 3 &
], Last][[-3 ;;]] // Column

Compile[] is also able to find common subexpressions if the whole expression is compilable in general. The resulting CompiledFunction contains a list of imperative program steps that is equal to the original expression.

expr = (1/(a^2 (b+m))) (-(-1+a+2 b) (-1+2 b+2 m)-
       4 b ArcTanh[1-2 b]+Log[4]+2 ((2+a) b+m+a m) Log[1+a]-
       a (b+m) Log[4-4 b]+a Log[(2-2 b)/(1+a)]-m Log[1-b]+
       2 Log[(1-b)/(1+a)]+(m-a (b+m)) Log[b]);

cf = Compile[{{a, _Real}, {b, _Real}, {m, _Real}}, Evaluate@expr]

Clicking on the + symbol reveals the generated code:

This expression can also be found as second to last part of the CompiledFunction expression:

cf[[-2]]

Function[{a, b, m}, 
 Block[{Compile`$5, Compile`$2, Compile`$21, Compile`$28, Compile`$33,
    Compile`$34}, Compile`$5 = -2 b; Compile`$2 = b + m; 
  Compile`$21 = 1 + a; Compile`$28 = 1/Compile`$21; Compile`$33 = -b; 
  Compile`$34 = 1 + Compile`$33; (1/(
  a^2 Compile`$2))((1 - a + Compile`$5) (-1 + 2 b + 2 m) - 
    4 b ArcTanh[1 + Compile`$5] + Log[4] + 
    2 ((2 + a) b + m + a m) Log[Compile`$21] - 
    a Compile`$2 Log[4 - 4 b] + a Log[Compile`$28 (2 + Compile`$5)] - 
    m Log[Compile`$34] + 
    2 Log[Compile`$28 Compile`$34] + (m - a Compile`$2) Log[b])]]