square roots don't simplify

[Akuli]

Here's the problem.

>>> from sympy import *
>>> from sympy.abc import x
>>> a=(x**2 + x*sqrt(x**2 + 1) + 1)/(x + sqrt(x**2 + 1))
>>> b=sqrt(x**2+1)
>>> solveset(Eq(a,b),x)        # these are equal
S.Complexes
>>> simplify(b)
sqrt(x**2 + 1)
>>> simplify(a)     # but this does not simplify to b
(x**2 + x*sqrt(x**2 + 1) + 1)/(x + sqrt(x**2 + 1))
>>> 

[smichr]

[cps]

If we let x**2 + 1 be y then a can be rewritten as (y + x*sqrt(y))/(x + sqrt(y)); factoring sqrt(y) from the numerator gives sqrt(y) (and that's valid because sqrt(y)*sqrt(y) is always y). I suspect that modifying factor to handle this sort of situation would be the way to go. I wonder if the fact that minpoly(a) gives _x**2 - (x**2 + 1) is a clue that "y = x**2 + 1" is the change of variables that will make this simplification more apparent... just pondering.

[avirlrma]

We'll have to improvise on some of the subfuctions that simplfy uses. I would love to work on this , just need some guidance from the maintainers.

[sanyamg123]

I would like to work on this issue. Please guide me.

[thedoctar]

This may not be a problem with the simplify function in and of itself, because sp.simplify(a/sp.sqrt(x**2+1)) gives 1 for me.

So when you give a hint to simplify about the common factor, it can factorise.

Maybe if you use more sophisticated algorithms using ring theory, you could solve this. But it would inevitably involve the algorithm having to guess possible factors.