\[ \int \frac {\tan ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\cos \left (d x +c \right )\right )}{a^{2} d}+\frac {\left (a^{2}-b^{2}\right )^{2} \left (5 a^{2}+b^{2}\right ) \ln \left (a +b \sec \left (d x +c \right )\right )}{a^{2} b^{6} d}-\frac {2 a \left (2 a^{2}-3 b^{2}\right ) \sec \left (d x +c \right )}{b^{5} d}+\frac {3 \left (a^{2}-b^{2}\right ) \left (\sec ^{2}\left (d x +c \right )\right )}{2 b^{4} d}-\frac {2 a \left (\sec ^{3}\left (d x +c \right )\right )}{3 b^{3} d}+\frac {\sec ^{4}\left (d x +c \right )}{4 b^{2} d}+\frac {\left (a^{2}-b^{2}\right )^{3}}{a \,b^{6} d \left (a +b \sec \left (d x +c \right )\right )} \]
command
integrate(tan(d*x+c)^7/(a+b*sec(d*x+c))^2,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________