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Well... The default Sage algorithms aren't always the most efficient.However, one can try :
sage: import sympy
sage: sympy.sympify((sqrt(2)+2)/(sqrt(2)+1)).simplify()._sage_()
sqrt(2)
We also have :
sage: ((sqrt(2)+2)/(sqrt(2)+1)/sqrt(2)).canonicalize_radical()
1
as well as :
sage: from giacpy_sage import *
sage: libgiac.simplify((sqrt(2)+2)/(sqrt(2)+1)).sage()
sqrt(2)
(this one can probably be done more economically...).
HTH...
