Automatically, we have (1-x^2) * sqrt(1-x^2) --> (1-x^2)^(3/2), but I don't
even
know of a command that crunches (4-4*x^2)*sqrt(1-x^2) --> 4 * (1-x^2)^
(3/2). Is there
a command that does this simplification?
OK:
(%i81) (1-x^2) * sqrt(1-x^2);
(%o81) (1-x^2)^(3/2)
Not so OK:
(%i87) (4-4*x^2) * sqrt(1-x^2);
(%o87) (4-4*x^2)*sqrt(1-x^2)
(%i88) ratsimp(%);
(%o88) (4-4*x^2)*sqrt(1-x^2)
(%i89) radcan(%);
(%o89) sqrt(1-x)*sqrt(x+1)*(4-4*x^2)
What I was really trying to do:
(%i98) hgfred([1,-1/2],[2],x^2) *x^2;
(%o98) (2*x^2*(sqrt(x^2)/(1-x^2)+((1-sqrt(1-x^2))*sqrt(x^2))/(1-x^2)^
(3/2)+(1-sqrt(1-x^2))/(sqrt(1-x^2)*sqrt(x^2)))*(1-x^2)^(3 /2))/(3*sqrt
(x^2))
Could be -(2/3) * (1-x^2)^(3/2) + constant:
(%i99) ratsimp(%);
(%o99) (sqrt(1-x^2)*(2*x^4-2*x^2)+2*x^2)/(3*x^2)
(%i100) factor(%);
(%o100) (2*(x^2*sqrt(1-x^2)-sqrt(1-x^2)+1))/3
Barton