\[ \int \frac {\cosh ^3(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (\cosh ^{2}\left (f x +e \right )\right ) \sinh \left (f x +e \right )}{3 a f \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}+\frac {2 \sinh \left (f x +e \right )}{3 a^{2} f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}} \]
command
integrate(cosh(f*x+e)^3/(a+b*sinh(f*x+e)^2)^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left ({\left (\frac {{\left (a^{3} e^{\left (12 \, e\right )} - 3 \, a b^{2} e^{\left (12 \, e\right )} + 2 \, b^{3} e^{\left (12 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}} + \frac {3 \, {\left (3 \, a^{3} e^{\left (10 \, e\right )} - 8 \, a^{2} b e^{\left (10 \, e\right )} + 7 \, a b^{2} e^{\left (10 \, e\right )} - 2 \, b^{3} e^{\left (10 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {3 \, {\left (3 \, a^{3} e^{\left (8 \, e\right )} - 8 \, a^{2} b e^{\left (8 \, e\right )} + 7 \, a b^{2} e^{\left (8 \, e\right )} - 2 \, b^{3} e^{\left (8 \, e\right )}\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}\right )} e^{\left (2 \, f x\right )} - \frac {a^{3} e^{\left (6 \, e\right )} - 3 \, a b^{2} e^{\left (6 \, e\right )} + 2 \, b^{3} e^{\left (6 \, e\right )}}{a^{4} e^{\left (6 \, e\right )} - 2 \, a^{3} b e^{\left (6 \, e\right )} + a^{2} b^{2} e^{\left (6 \, e\right )}}}{3 \, {\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac {3}{2}} f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________