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