inconsistency in simplify_radical

[wdjoyner]

This is a problem:

sage: f = sqrt(25-x)*sqrt(1+1/(4*(25-x)))
sage: f.integral(x,9,16)
integrate(sqrt(1/(4*(25 - x)) + 1)*sqrt(25 - x), x, 9, 16)
sage: f.nintegral(x,9,16)
(24.9153783348643, 2.7661626694613149e-13, 21, 0)
sage: g = f.simplify_radical()
sage: g.integral(x,9,16)
I*(65*sqrt(65)*I/6 - 37*sqrt(37)*I/6)/2
sage: ans = g.integral(x,9,16)
sage: ans.real()
(37*sqrt(37)/6 - 65*sqrt(65)/6)/2
sage: RR(ans.real())
-24.9153783348643

CC: @burcin @aghitza @orlitzky

Component: symbolics

Reviewer: Michael Orlitzky, Karl-Dieter Crisman

Issue created by migration from https://trac.sagemath.org/ticket/3520

[haraldschilly]

comment:1

Just for reference, the same with MMA6:

In[3]:=
f[x_] := Sqrt[25 - x]*Sqrt[1 + 1/(4*(25 - x))]

In[16]:=
g[x_] := Integrate[f[x], x]; FullSimplify[g[x]]

Out[16]=
(-(1/12))*(4 + 1/(25 - x))^(3/2)*(25 - x)^(3/2)

In[15]:=
FullSimplify[D[g[x], x] - f[x]]

Out[15]=
0

In[6]:=
Integrate[f[x], {x, 9, 16}]

Out[6]=
(1/12)*(-37*Sqrt[37] + 65*Sqrt[65])

In[8]:=
NIntegrate[f[x], {x, 9, 16}, WorkingPrecision -> 40]

Out[8]=
24.915378334864299909236795241439053518882655529789`40.

[kcrisman]

comment:2

With the new symbolics and Maxima 5.19.1:

sage: sqrt(25-x)*sqrt(1+1/(4*(25-x)))
sqrt(-1/4/(x - 25) + 1)*sqrt(-x + 25)
sage: f = _
sage: f.integral(x)
1/12*(4*I*x - 101*I)*sqrt(4*x - 101)
sage: f.integral(x,9,16)
-37/12*sqrt(37) + 65/12*sqrt(65)
sage: f.nintegral(x,9,16)
(24.9153783348643, 2.7661626694613149e-13, 21, 0)
sage: g = f.simplify_radical()
sage: g.integral(x,9,16)
37/12*sqrt(37) - 65/12*sqrt(65)
sage: ans = g.integral(x,9,16)
sage: ans.real()
37/12*sqrt(37) - 65/12*sqrt(65)
sage: RR(ans.real())
-24.9153783348643

Maxima can now integrate the original one, but still gives the wrong simplification (?) of f. It seems to be choosing the wrong square root of negative one, as it were, since

sage: j = -g
sage: j.integrate(x,9,16)
-37/12*sqrt(37) + 65/12*sqrt(65)

So the real problem is in simplify_radical().

[kcrisman]

comment:4

One may want to check the latest CVS of Maxima, where I believe some definitions relating to this have changed.

[kcrisman]

comment:5

Just updating. This is still in 5.20.1, but it's not clear whether this is really a bug, since radical simplifications will in their nature sometimes change which square root of -1 they use, and perhaps it's not possible to do so consistently across multiplication or division. Comments?

[kcrisman]

comment:6

Maxima devs have been discussing some things related to this, so it could be worth checking whether this has changed again in their CVS.

[kcrisman]

comment:7

What is really going on here is that simplify_radical uses radcan under the hood, which treats things as symbolic expressions, not functions, and just chooses a branch - consistently, but arbitrarily. At least I think that is what is here. See the Maxima list thread starting here.