\[ \int \frac {1}{\left (d+e x^2\right ) \left (d^2-e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{4 d^{2} \left (e \,x^{2}+d \right )}+\frac {\arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{2 d^{\frac {5}{2}} \sqrt {e}}+\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{4 d^{\frac {5}{2}} \sqrt {e}} \]
command
integrate(1/(e*x^2+d)/(-e^2*x^4+d^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {5}{2}}} - \frac {\arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{4 \, \sqrt {-d e} d^{2}} + \frac {x}{4 \, {\left (x^{2} e + d\right )} d^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________