@Markiyan Hirnyk:
Digits:= 100:

Thank you, Markiyan Hirnyk - the problem is completely resolved.
Thank other conversation participants for their help! (Carl Love and nm)

The clue was in accuracy: when I changed the "Digits" parameter to 30 the function f(x) (which in it's turn is a Maple computation result) changed a lot. What's more, now
int(f(x), x=0..2) = 6.08579120223268572952318143285,

what is the same as the "some other computer algebra system"'s

                           6.0857912022

If it may be of use to you, the file with amendments:

Challenge_02_resolve.mw

P.S. "We are as good as our words and our words are as good as we are."
Now we can enjoy the ammended f(x) plot   (in file it is called g(x))

The function was cut in a half after I've posted it. The file itself.

There are several problems in the file:

1. Integrals calculated via different approaches give different results. (especially the last one)

2. Plots of the same function f(x) are not identical.

3. If you restart the file in a new engine (open in the next tab in Maple) the results may vary dramatically.

So, 

what is the way to obtain robust integral estimations?

 Windows 8 (64-bit), Maple 17.00

Challenge_01.mw