"Smart" substitution of subexpressions

I have the following question.

An expression, which I want to simplify contains several subexpressions which appear quite frequently all over the place. To optimize simplification I would like to use abbreviations for some of them. Is there any way to do it in a "smart" way, i.e. to account for subexpressions which differ only by sign/multiplication by a number or a variable? Here is an example to illustrate what I mean.

For example, the adverted subexpression is:

-a^2 + b^2/(c^2 - d^2)

and I want to use variable A1 everywhere instead it:

-a^2 + b^2/(c^2 - d^2) -> A1

Now, I want Mathematica to substitute the expressions which are essentially equal to this one, but are simply written in another form like:

-a^2 - b^2/(d^2 - c^2)
-a^2 + (-b^2/(d^2 - c^2))

Also it would be great to use this rule for expressions like

-2*a^2 + 2*b^2/(c^2 - d^2) (*2*A1*)

or

a^2 - b^2/(c^2 - d^2) (*-A1*)

or even

-x*a^2 + x*b^2/(c^2 - d^2) (*x*A1*)

Is there a way to do it?

Short version :

When one wants to do f[a+b] /. a+b->c, it is often more efficient to write f[a+b] /. a-> c-b and simplify the result ( with Simplify, Expand...).

Long version :

You can apply the rule b -> -Sqrt[a^2 + A1] Sqrt[c^2 - d^2] (equivalent to -a^2 + b^2/(c^2 - d^2) -> A1) and afterward try to simplify.

In fact, your example is a little bit more complex because there are to possible rules b-> Sqrt[...] and b-> -Sqrt[...], but it works fine :

rule = Solve[-a^2 + b^2/(c^2 - d^2) == A1, b]

transfomation[x_] := x /. rule // ExpandAll // Together

-a^2 - b^2/(d^2 - c^2) // transfomation
-a^2 + (-b^2/(d^2 - c^2)) // transfomation
-2*a^2 + 2*b^2/(c^2 - d^2) (*2*A1*) // transfomation
a^2 - b^2/(c^2 - d^2) (*-A1*)// transfomation
-x*a^2 + x*b^2/(c^2 - d^2) (*x*A1*)  // transfomation

{{b -> -Sqrt[a^2 + A1] Sqrt[c^2 - d^2]}, {b -> Sqrt[a^2 + A1] Sqrt[c^2 - d^2]}}

{A1, A1}

{A1, A1}

{2 A1, 2 A1}

{-A1, -A1}

{A1 x, A1 x}

• Thanks a lot for your answer, but I'm afraid it is not precisely what I needed. Yes, it works in these cases but I also would like it to leave b variable as it is when it does not appear in the combination: -a^2 + b^2/(c^2 - d^2) For example: b/(-a^2 + b^2/(c^2 - d^2)) = b/A1. Secondly, this subexpression was chosen just as an illustration. I think, this method might not work when it is hard (if possible) to express one of the variables through the others. Jan 5 '19 at 21:54
• I understand, but I have nothing better to propose (These kinds of apparently trivial algebric manipulations are often very frustating). Jan 5 '19 at 22:00
• I you want, you can add further more complicated examples in your question. Generally speaking, it is not recommended to change the question, but as I'm the only one who has given a answer I can delete it. (I don't mind the +50 of reputation) Jan 5 '19 at 22:34