\[ \int \frac {\text {csch}^2(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx \]
Optimal antiderivative \[ -\frac {\coth \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{a f}-\frac {\sqrt {2}\, \sqrt {\frac {1}{1+\cosh \left (2 f x +2 e \right )}}\, \sqrt {1+\sinh ^{2}\left (f x +e \right )}\, \EllipticE \left (\frac {\sinh \left (f x +e \right )}{\sqrt {1+\sinh ^{2}\left (f x +e \right )}}, \sqrt {1-\frac {b}{a}}\right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{a f \sqrt {\frac {\mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )}{a}}}+\frac {\sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \tanh \left (f x +e \right )}{a f} \]
command
integrate(csch(f*x+e)^2/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, {\left (\frac {\arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} - \sqrt {b}}{2 \, \sqrt {-a}}\right ) e^{\left (-2 \, e\right )}}{\sqrt {-a}} - \frac {2 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} + \sqrt {b}\right )} e^{\left (-2 \, e\right )}}{{\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} - 2 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} \sqrt {b} - 4 \, a + b}\right )} e^{\left (3 \, e\right )}}{f^{2}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________