\[ \int \frac {\text {csch}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {\cosh \left (f x +e \right ) \sqrt {a}}{\sqrt {a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )}}\right )}{a^{\frac {5}{2}} f}-\frac {b \cosh \left (f x +e \right )}{3 a \left (a -b \right ) f \left (a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}}}-\frac {\left (5 a -3 b \right ) b \cosh \left (f x +e \right )}{3 a^{2} \left (a -b \right )^{2} f \sqrt {a -b +b \left (\cosh ^{2}\left (f x +e \right )\right )}} \]
command
integrate(csch(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
\[ -\frac {{\left (\frac {{\left ({\left (\frac {{\left (5 \, a^{9} b^{2} e^{\left (12 \, e\right )} - 3 \, a^{8} b^{3} e^{\left (12 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}} + \frac {3 \, {\left (8 \, a^{10} b e^{\left (10 \, e\right )} - 7 \, a^{9} b^{2} e^{\left (10 \, e\right )} + a^{8} b^{3} e^{\left (10 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} + \frac {3 \, {\left (8 \, a^{10} b e^{\left (8 \, e\right )} - 7 \, a^{9} b^{2} e^{\left (8 \, e\right )} + a^{8} b^{3} e^{\left (8 \, e\right )}\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}\right )} e^{\left (2 \, f x\right )} + \frac {5 \, a^{9} b^{2} e^{\left (6 \, e\right )} - 3 \, a^{8} b^{3} e^{\left (6 \, e\right )}}{a^{12} e^{\left (12 \, e\right )} - 2 \, a^{11} b e^{\left (12 \, e\right )} + a^{10} b^{2} e^{\left (12 \, e\right )}}}{{\left (b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b\right )}^{\frac {3}{2}}} - \frac {6 \, \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right ) e^{\left (-6 \, e\right )}}{\sqrt {-a} a^{2}}\right )} e^{\left (6 \, e\right )}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________