\[ \int \frac {\sqrt {x}}{\sqrt {1+x} \left (1+x^2\right )} \, dx \]
Optimal antiderivative \[ -\frac {\left (1-i\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {1-i}\, \sqrt {x}}{\sqrt {1+x}}\right )}{2}-\frac {\left (1+i\right )^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {1+i}\, \sqrt {x}}{\sqrt {1+x}}\right )}{2} \]
command
integrate(x^(1/2)/(x^2+1)/(1+x)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) + \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Timed out} \]________________________________________________________________________________________