\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+a \sec (e+f x)} \, dx \]
Optimal antiderivative \[ \frac {3 d \left (2 c^{2}-2 c d +d^{2}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a f}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{f \left (a +a \sec \left (f x +e \right )\right )}-\frac {d \left (4 c^{2}-12 c d +4 d^{2}+\left (2 c -3 d \right ) d \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{2 a f} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^3/(a+a*sec(f*x+e)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a} - \frac {3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{a} - \frac {2 \, {\left (6 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a}}{2 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________