\[ \int \sec ^5(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {\left (48 a^{2}+80 a b +35 b^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{128 f}+\frac {\left (48 a^{2}+80 a b +35 b^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{128 f}+\frac {\left (48 a^{2}+80 a b +35 b^{2}\right ) \left (\sec ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{192 f}+\frac {b \left (10 a +7 b \right ) \left (\sec ^{5}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{48 f}+\frac {b \left (\sec ^{7}\left (f x +e \right )\right ) \left (a +b -a \left (\sin ^{2}\left (f x +e \right )\right )\right ) \tan \left (f x +e \right )}{8 f} \]
command
integrate(sec(f*x+e)^5*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 3 \, {\left (48 \, a^{2} + 80 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (144 \, a^{2} \sin \left (f x + e\right )^{7} + 240 \, a b \sin \left (f x + e\right )^{7} + 105 \, b^{2} \sin \left (f x + e\right )^{7} - 528 \, a^{2} \sin \left (f x + e\right )^{5} - 880 \, a b \sin \left (f x + e\right )^{5} - 385 \, b^{2} \sin \left (f x + e\right )^{5} + 624 \, a^{2} \sin \left (f x + e\right )^{3} + 1168 \, a b \sin \left (f x + e\right )^{3} + 511 \, b^{2} \sin \left (f x + e\right )^{3} - 240 \, a^{2} \sin \left (f x + e\right ) - 528 \, a b \sin \left (f x + e\right ) - 279 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{4}}}{768 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________