\[ \int \frac {\sinh ^6(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {a \cosh \left (f x +e \right ) \left (\sinh ^{3}\left (f x +e \right )\right )}{\left (a -b \right ) b f \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}+\frac {\left (4 a -b \right ) \cosh \left (f x +e \right ) \sinh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{3 \left (a -b \right ) b^{2} f}+\frac {\left (8 a^{2}-3 a b -2 b^{2}\right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cosh \left (2 f x +2 e \right )}}\, \sqrt {1+\sinh ^{2}\left (f x +e \right )}\, \EllipticE \left (\frac {\sinh \left (f x +e \right )}{\sqrt {1+\sinh ^{2}\left (f x +e \right )}}, \sqrt {1-\frac {b}{a}}\right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{3 \left (a -b \right ) b^{3} f \sqrt {\frac {\mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )}{a}}}-\frac {\left (4 a -b \right ) \sqrt {2}\, \sqrt {\frac {1}{1+\cosh \left (2 f x +2 e \right )}}\, \sqrt {1+\sinh ^{2}\left (f x +e \right )}\, \EllipticF \left (\frac {\sinh \left (f x +e \right )}{\sqrt {1+\sinh ^{2}\left (f x +e \right )}}, \sqrt {1-\frac {b}{a}}\right ) \mathrm {sech}\left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}}{3 \left (a -b \right ) b^{2} f \sqrt {\frac {\mathrm {sech}\left (f x +e \right )^{2} \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right )}{a}}}-\frac {\left (8 a^{2}-3 a b -2 b^{2}\right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, \tanh \left (f x +e \right )}{3 \left (a -b \right ) b^{3} f} \]
command
integrate(sinh(f*x+e)^6/(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________