\[ \int \sec ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{16 f}+\frac {\left (8 a^{2}+12 a b +5 b^{2}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}+\frac {b \left (8 a +5 b \right ) \left (\sec ^{3}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{24 f}+\frac {b \left (\sec ^{5}\left (f x +e \right )\right ) \left (a +b -a \left (\sin ^{2}\left (f x +e \right )\right )\right ) \tan \left (f x +e \right )}{6 f} \]
command
integrate(sec(f*x+e)^3*(a+b*sec(f*x+e)^2)^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) - 3 \, {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (24 \, a^{2} \sin \left (f x + e\right )^{5} + 36 \, a b \sin \left (f x + e\right )^{5} + 15 \, b^{2} \sin \left (f x + e\right )^{5} - 48 \, a^{2} \sin \left (f x + e\right )^{3} - 96 \, a b \sin \left (f x + e\right )^{3} - 40 \, b^{2} \sin \left (f x + e\right )^{3} + 24 \, a^{2} \sin \left (f x + e\right ) + 60 \, a b \sin \left (f x + e\right ) + 33 \, b^{2} \sin \left (f x + e\right )\right )}}{{\left (\sin \left (f x + e\right )^{2} - 1\right )}^{3}}}{96 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________