{\displaystyle
\int\frac{x^2\hbox{d}x}{(1+x^2)\sqrt{1-x^2}}=\\
\int\frac{1+x^2-1}{(1+x^2)\sqrt{1-x^2}}\hbox{d}x=\\
\int\frac{\hbox{d}x}{\sqrt{1-x^2}}-\int\frac{\hbox{d}x}{(1+x^2)\sqrt{1-x^2}}}