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