\[ \int \frac {\sec (e+f x) (c-c \sec (e+f x))^4}{a+a \sec (e+f x)} \, dx \]
Optimal antiderivative \[ -\frac {35 c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{2 a f}+\frac {28 c^{4} \tan \left (f x +e \right )}{a f}-\frac {21 c^{4} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 a f}+\frac {2 c \left (c -c \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{f \left (a +a \sec \left (f x +e \right )\right )}+\frac {7 c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f} \]
command
integrate(sec(f*x+e)*(c-c*sec(f*x+e))^4/(a+a*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a} - \frac {105 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a} - \frac {96 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a} + \frac {2 \, {\left (87 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 136 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 57 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________