How do symbolic math software work? As the answer to a question of mine I was referred to a website (see here please)
How can WolframAlpha do it like humans?

For the specific system (Mathematica) that you mentioned, there are  descriptions of its internals on these web pages and these. But Mathematica is a commercial system, so its internal workings are proprietary, which is why the descriptions don't provide much detail.
While these systems might appear to work "the same way as a human", they really don't. Human beings often use clever creative tricks to solve problems. Computers in general (and computer algebra systems in particular) typically use brute-force systematic methods, as described in the referenced materials. See the description of how Mathematica finds indefinite integrals, for example.
The best results are obtained by a combination of a powerful brute-force computing and intelligent guidance provided by a human being. Computer algebra systems are enormously useful, but they only do what you tell them to do  :-)
