\[ \int \cos ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {b^{2} \arctanh \left (\sin \left (f x +e \right )\right )}{f}+\frac {a \left (a +2 b \right ) \sin \left (f x +e \right )}{f}-\frac {a^{2} \left (\sin ^{3}\left (f x +e \right )\right )}{3 f} \]
command
integrate(cos(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
\[ -\frac {2 \, a^{2} \sin \left (f x + e\right )^{3} - 3 \, b^{2} \log \left ({\left | \sin \left (f x + e\right ) + 1 \right |}\right ) + 3 \, b^{2} \log \left ({\left | \sin \left (f x + e\right ) - 1 \right |}\right ) - 6 \, a^{2} \sin \left (f x + e\right ) - 12 \, a b \sin \left (f x + e\right )}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________