\[ \int \cos ^7(c+d x) (a+b \tan (c+d x))^3 \, dx \]
Optimal antiderivative \[ \frac {8 a \left (2 a^{2}+b^{2}\right ) \sin \left (d x +c \right )}{35 d}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right ) \left (b -2 a \tan \left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}{35 d}+\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}{7 d}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \left (b \left (6 a^{2}+b^{2}\right )-a \left (4 a^{2}-b^{2}\right ) \tan \left (d x +c \right )\right )}{35 d} \]
command
integrate(cos(d*x+c)^7*(a+b*tan(d*x+c))^3,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 \[ \text {Timed out} \]_______________________________________________________