\[ \int \frac {1}{\sqrt {1+x^2} \sqrt {2+2 x^2}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \! \left (x \right ) \sqrt {2}}{2} \]
command
integrate(1/(x^2+1)^(1/2)/(2*x^2+2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {1}{4} \, \sqrt {2} i \log \left (i x - 1\right ) - \frac {1}{4} \, \sqrt {2} i \log \left (-i x - 1\right ) \]