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