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