\[ \int \frac {1}{\left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {x}{8 d^{2} \left (e \,x^{2}+d \right )^{2}}+\frac {5 x}{16 d^{3} \left (e \,x^{2}+d \right )}+\frac {7 \arctan \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{16 d^{\frac {7}{2}} \sqrt {e}}+\frac {\arctanh \left (\frac {x \sqrt {e}}{\sqrt {d}}\right )}{8 d^{\frac {7}{2}} \sqrt {e}} \]
command
integrate(1/(e*x^2+d)^2/(-e^2*x^4+d^2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {7 \, \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{16 \, d^{\frac {7}{2}}} - \frac {\arctan \left (\frac {x e}{\sqrt {-d e}}\right )}{8 \, \sqrt {-d e} d^{3}} + \frac {5 \, x^{3} e + 7 \, d x}{16 \, {\left (x^{2} e + d\right )}^{2} d^{3}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________