\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^3}{c+d \sec (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {a^{3} \arctanh \left (\sin \left (f x +e \right )\right )}{2 d f}+\frac {a^{3} \left (c^{2}-3 c d +3 d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{d^{3} f}-\frac {2 a^{3} \left (c -d \right )^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {c -d}\, \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {c +d}}\right )}{d^{3} f \sqrt {c +d}}-\frac {a^{3} \left (c -3 d \right ) \tan \left (f x +e \right )}{d^{2} f}+\frac {a^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 d f} \]
command
integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3/(c+d*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{d^{3}} - \frac {{\left (2 \, a^{3} c^{2} - 6 \, a^{3} c d + 7 \, a^{3} d^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{d^{3}} + \frac {4 \, {\left (a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 \, a^{3} c d^{2} - a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{\sqrt {-c^{2} + d^{2}} d^{3}} + \frac {2 \, {\left (2 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, a^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} d^{2}}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________