\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^6}{(a+a \sec (e+f x))^3} \, dx \]
Optimal antiderivative \[ \frac {d^{3} \left (40 c^{3}-90 c^{2} d +78 c \,d^{2}-23 d^{3}\right ) \arctanh \left (\sin \left (f x +e \right )\right )}{2 a^{3} f}-\frac {2 d \left (2 c^{5}+18 c^{4} d +107 c^{3} d^{2}-472 c^{2} d^{3}+456 c \,d^{4}-136 d^{5}\right ) \tan \left (f x +e \right )}{15 a^{3} f}-\frac {d^{2} \left (4 c^{4}+36 c^{3} d +216 c^{2} d^{2}-626 c \,d^{3}+345 d^{4}\right ) \sec \left (f x +e \right ) \tan \left (f x +e \right )}{30 a^{3} f}-\frac {d \left (2 c^{3}+18 c^{2} d +111 c \,d^{2}-136 d^{3}\right ) \left (c +d \sec \left (f x +e \right )\right )^{2} \tan \left (f x +e \right )}{15 a^{3} f}+\frac {\left (c -d \right ) \left (2 c^{2}+18 c d +115 d^{2}\right ) \left (c +d \sec \left (f x +e \right )\right )^{3} \tan \left (f x +e \right )}{15 f \left (a^{3}+a^{3} \sec \left (f x +e \right )\right )}+\frac {\left (c -d \right ) \left (2 c +13 d \right ) \left (c +d \sec \left (f x +e \right )\right )^{4} \tan \left (f x +e \right )}{15 a f \left (a +a \sec \left (f x +e \right )\right )^{2}}+\frac {\left (c -d \right ) \left (c +d \sec \left (f x +e \right )\right )^{5} \tan \left (f x +e \right )}{5 f \left (a +a \sec \left (f x +e \right )\right )^{3}} \]
command
integrate(sec(f*x+e)*(c+d*sec(f*x+e))^6/(a+a*sec(f*x+e))^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {30 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{a^{3}} - \frac {30 \, {\left (40 \, c^{3} d^{3} - 90 \, c^{2} d^{4} + 78 \, c d^{5} - 23 \, d^{6}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{a^{3}} - \frac {20 \, {\left (90 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 126 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 51 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 180 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 216 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 76 \, d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 90 \, c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 90 \, c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 33 \, d^{6} \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 )}^{3} a^{3}} + \frac {3 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{12} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 45 \, a^{12} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 60 \, a^{12} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 45 \, a^{12} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{12} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a^{12} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 10 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 150 \, a^{12} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 400 \, a^{12} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 450 \, a^{12} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 240 \, a^{12} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 50 \, a^{12} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{12} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 90 \, a^{12} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 225 \, a^{12} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2100 \, a^{12} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3825 \, a^{12} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2790 \, a^{12} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 735 \, a^{12} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{15}}}{60 \, f} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________