\[ \int \cos ^6(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx \]
Optimal antiderivative \[ \frac {\left (5 a^{2}+2 a b +b^{2}\right ) x}{16}+\frac {\left (5 a^{2}+2 a b +b^{2}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )}{16 d}+\frac {\left (a -b \right ) \left (5 a +3 b \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{24 d}+\frac {\left (a -b \right ) \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \left (a +b \left (\tan ^{2}\left (d x +c \right )\right )\right )}{6 d} \]
command
integrate(cos(d*x+c)^6*(a+b*tan(d*x+c)^2)^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 \[ \text {Timed out} \]_______________________________________________________