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How common is it for researchers to use computer algebra systems? I'm working in electrical engineering and moving into quite theoretical areas. For example, I'm reading right now some papers on tail bounds / concentration of measure in probability. They pull almost out of nowhere some very clever functions that make the inequalities work in a nice way.

For example, they write

$\frac{b}{b - a}e^{ta} - \frac{a}{b - a}e^{tb} = e^{g(u)}$

Where $g(u) = -\gamma u + ln(1 - \gamma + \gamma e^u)$, $u = t(b - a)$, $\gamma = \frac{-a}{b - a}$

This all works out nicely, but it would be certainly much harder to go backwards. This is also a very simple example compared to others I've seen.

I can understand coming up with this through a trial and error process. Or perhaps they are using standard tricks I'm just not aware of yet. But, how can one know that such a function $g(u)$ even exists, especially in more general situations?

Is it possible to use a CAS and ask it "write this function in the form $e^{g(u)}$, or tell me it's not possible"? Is a CAS useful in general? Is it common to make use of them?

RJTK
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    It's getting more common all the time. The thing is, you have to know the right question to ask. The answer to "write in the form $\exp(g(u))$" is $g(u) = \log (\text{})$. You may or may not be able to simplify that log to something nice. Unless you can tell the CAS the form you want, a CAS might not be helpful: it might find some simplification, but it can't read your mind so it might not be the "right" simplification, i.e. the one you want. – Robert Israel Jun 22 '16 at 19:59
  • Right, maybe the way I am asking the question is kind of dumb. I hadn't put in too much thought before asking... Perhaps I should just ask, do people use a CAS to help get these kinds of clever simplifications, or is it just through a ton of trial and error, or "seeing" some kind of nice structure in the equations? – RJTK Jun 22 '16 at 20:10

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