\[ \int \frac {\sinh (e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\cosh \left (f x +e \right )}{3 \left (a -b \right ) f \left (a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \cosh \left (f x +e \right )}{3 \left (a -b \right )^{2} f \sqrt {a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )}} \]
command
integrate(sinh(f*x+e)/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (\frac {a^{2} b e^{\left (6 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}} + {\left ({\left (\frac {a^{2} b e^{\left (2 \, f x + 12 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}} + \frac {3 \, {\left (2 \, a^{3} e^{\left (10 \, e\right )} - a^{2} b e^{\left (10 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )} + \frac {3 \, {\left (2 \, a^{3} e^{\left (8 \, e\right )} - a^{2} b e^{\left (8 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )}\right )}}{3 \, {\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac {3}{2}} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________