\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx \]
Optimal antiderivative \[ \frac {\left (3 c -2 d \right ) d^{2} \arctanh \left (\sin \left (f x +e \right )\right )}{a^{2} f}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{3 f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {\left (c^{3}+4 c^{2} d -12 c \,d^{2}+10 d^{3}-\left (c -4 d \right ) d^{2} \sec \left (f x +e \right )\right ) \tan \left (f x +e \right )}{3 f \left (a^{2}+a^{2} \sec \left (f x +e \right )\right )} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^3/(a+a*sec(f*x+e))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {12 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} a^{2}} - \frac {6 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac {6 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}} + \frac {a^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{4} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - a^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 9 \, a^{4} c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 27 \, a^{4} c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6}}}{6 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________