As I reported maxima gives wrong answer on integrands like
x^n/sqrt(1-x^2)/(1+a^2*x^2)^m with even powers n,m.
Integrands like asin(x)*x/(1+a^2*x^2)^2 are in close connection with them.
Maxima fails on it too. You can substitute acos in place of asin
(C7) factor(defint(2*x*asin(x)/(1+x^2)^2,x,-1,1));
%PI
(D7) - --- (lose contibution sqrt(2)/2)
2
(C8) factor(defint(2*x*acos(x)/(1+x^2)^2,x,-1,1));
%PI
(D8) --- (lose contibution -sqrt(2)/2)
2
(C9) factor(defint(2*x*atan(x)/(1+x^2)^2,x,-1,1));
1
(D9) -
2
The last one is correct. Probably due different procedure to integrate (by parts)
However
(C10) factor(defint(2*x*acot(x)/(1+x^2)^2,x,-1,1));
1
(D10) - (lose contibution -%pi/2)
2
rgds,
v