This report comes from the Sage support list thread at http://groups.google.com/group/sage-support/browse_thread/thread/5075ee447ee6fdae/e1cb3b38f38e80c9

The fact that the first and second are different isn't good; probably the first one is an oversimplification by radcan?

With maxima 5.23.2:

(%i1) radcan((2*x^2/(x^2+1)-1)/sqrt(-4*x^2/(x^2+1)^2+1) );

(%o1)

1

radcan(((2*x^2-x^2-1)/(x^2+1))/sqrt((x^2-1)^2/(x^2+1)^2));

2

x - 1

(%o2)

--------

!

2 !

!

x - 1!

radcan(((x^2-1)*x/(x^2+1)^2-x/(x^2+1))/sqrt(-(x^2-1)^2/(x^2 +

1)^2 + 1) );

x

(%o3)

- ---------------

2

(x + 1) abs(x)