Re: sqrt(1-x^2)/(2-2*x^2) --/--> 1/(2*sqrt(1-x^2))

Maxima calls inte to do integrate((sqrt(1-x^2)*(x^2+1))/(2-2*x^2),x,0,1). If I recall, 
inte is one of the last functions integrate tries. Some other method (not sure which one) 
handles integrate((x^2+1)/(2*sqrt(1-x^2)),x,0,1). 

integrate((sqrt(1-x^2)*(x^2+1))/(2-2*x^2),x,0,1) is in the testsuite.

--Barton