\[ \int \frac {(c-c \sec (e+f x))^4}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {c^{4} x}{a^{2}}-\frac {6 c^{4} \arctanh \left (\sin \left (f x +e \right )\right )}{a^{2} f}-\frac {16 c^{4} \cot \left (f x +e \right )}{a^{2} f}-\frac {32 c^{4} \left (\cot ^{3}\left (f x +e \right )\right )}{3 a^{2} f}+\frac {32 c^{4} \left (\csc ^{3}\left (f x +e \right )\right )}{3 a^{2} f}+\frac {c^{4} \tan \left (f x +e \right )}{a^{2} f} \]
command
integrate((c-c*sec(f*x+e))^4/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, {\left (f x + e\right )} c^{4}}{a^{2}} - \frac {18 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {18 \, c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} - \frac {6 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2}} + \frac {8 \, {\left (a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a^{6}}}{3 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________