\[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sinh \left (f x +e \right ) \sqrt {a -b}}{\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}\right )}{\left (a -b \right )^{\frac {3}{2}} f}-\frac {b \sinh \left (f x +e \right )}{a \left (a -b \right ) f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sech(f*x+e)/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\frac {\frac {{\left (a^{2} b e^{\left (4 \, e\right )} - 2 \, a b^{2} e^{\left (4 \, e\right )} + b^{3} e^{\left (4 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, e\right )}} - \frac {a^{2} b e^{\left (2 \, e\right )} - 2 \, a b^{2} e^{\left (2 \, e\right )} + b^{3} e^{\left (2 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 3 \, a^{3} b e^{\left (6 \, e\right )} + 3 \, a^{2} b^{2} e^{\left (6 \, e\right )} - a b^{3} e^{\left (6 \, e\right )}}}{\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}} - \frac {2 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} + \sqrt {b}}{2 \, \sqrt {a - b}}\right )}{{\left (a e^{\left (4 \, e\right )} - b e^{\left (4 \, e\right )}\right )} \sqrt {a - b}}\right )} e^{\left (4 \, e\right )}}{f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________