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Great Q. I use Mathematica when I want to perform some very complicated symbolic-type operation to test some hypothesis I have. Or, in one case, to assist in a proof (see http://math.mit.edu/~urschel/instabilities.pdf …). One of the biggest benefits of WolframAlpha for me is interface and ease of use.
there are many benefits to the WA interface if you just want to get a rough idea of something and you don't want to specify, say, the range of a plot .. I also thought that MM had a built in WA interface via ctrl+= command, but it seems like the added benefit 1/2
from what i've been able to read is that you can use the interface and its output as input to other calculations .. For heavy lifting it's probably not the go-to but for students and academics looking for a quick rough fix i can see the benefit
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