\[ \int x \sqrt {\frac {1-x^2}{1+x^2}} \, dx \]
Optimal antiderivative \[ -\arctan \! \left (\sqrt {\frac {-x^{2}+1}{x^{2}+1}}\right )+\frac {\left (x^{2}+1\right ) \sqrt {\frac {-x^{2}+1}{x^{2}+1}}}{2} \]
command
integrate(x*((-x**2+1)/(x**2+1))**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \int x \sqrt {- \frac {\left (x - 1\right ) \left (x + 1\right )}{x^{2} + 1}}\, dx \]
Sympy 1.8 under Python 3.8.8 output
\[ \begin {cases} \frac {\sqrt {1 - x^{2}} \sqrt {x^{2} + 1}}{2} - \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {1 - x^{2}}}{2} \right )} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]