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If I do

Simplify[(c^3 - s^3)^2 - (s^3 + c^3)^2, 
 TransformationFunctions -> Automatic]

(i.e. a verbose version of default behaviour), what transformation functions actually get applied? Similarly for FullSimplify.

The documentation says By default FullSimplify does not use Reduce:. Well how would I know? What else doesn't it use? What does it use?

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    $\begingroup$ @NasserM.Abbasi LeafCount is the ComplexityFunction not the TransformationFunction. The question is about the latter. $\endgroup$
    – Sjoerd C. de Vries
    Dec 7 '12 at 16:42
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    $\begingroup$ +1 because I like questions about the infrastructure, but honestly I think this cannot possibly be answered. These are the proprietary inner workings of Mathematica and they aren't going to be shared freely, in my estimation. $\endgroup$
    – Mr.Wizard
    Dec 7 '12 at 18:32
  • $\begingroup$ Its only a list of functions though right? No big secret I would have thought $\endgroup$
    – WalkingRandomly
    Dec 7 '12 at 22:51
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The documentation for FullSimplify lists the following transformations in the section Examples → Properties & Relations (I do not think this the list is complete): Expand, TrigExpand, PiecewiseExpand, FunctionExpand, LogicalExpand, Factor, FactorSquareFree, TrigFactor, RootReduce, ToRadicals, Together, Apart.

It also mentions that

PowerExpand makes special assumptions on input and is not used by FullSimplify ComplexExpand assumes variables to be real and is also not used by FullSimplify

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    $\begingroup$ This is potentially quite useful information: if FullSimplify is not giving the desired form, it makes sense to try PowerExpand and ComplexExpand manually. $\endgroup$
    – bill s
    Jun 23 '13 at 5:08
  • $\begingroup$ I very often invoke ComplexExpand manually before FullSimplify, sometimes providing different TargetFunctions option to get desired result. $\endgroup$
    – Vladimir Reshetnikov
    Jun 23 '13 at 15:42

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