\[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {c^{4} x}{a^{3}}+\frac {c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{a^{3} f}-\frac {3 c^{4} \tan \left (f x +e \right )}{a^{3} f \left (1+\sec \left (f x +e \right )\right )^{3}}-\frac {c^{4} \left (\sec ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{5 a^{3} f \left (1+\sec \left (f x +e \right )\right )^{3}}+\frac {14 c^{4} \tan \left (f x +e \right )}{5 a^{3} f \left (1+\sec \left (f x +e \right )\right )^{2}}-\frac {23 c^{4} \tan \left (f x +e \right )}{5 a^{3} f \left (1+\sec \left (f x +e \right )\right )} \]
command
integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {5 \, {\left (f x + e\right )} c^{4}}{a^{3}} + \frac {5 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {5 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 5 \, a^{12} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{15}}}{5 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________