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