\[ \int \cos ^7(c+d x) (a+b \tan (c+d x))^2 \, dx \]
Optimal antiderivative \[ -\frac {5 a b \left (\cos ^{7}\left (d x +c \right )\right )}{42 d}+\frac {\left (6 a^{2}+b^{2}\right ) \sin \left (d x +c \right )}{6 d}-\frac {\left (6 a^{2}+b^{2}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{6 d}+\frac {\left (6 a^{2}+b^{2}\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{10 d}-\frac {\left (6 a^{2}+b^{2}\right ) \left (\sin ^{7}\left (d x +c \right )\right )}{42 d}-\frac {b \left (\cos ^{7}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )}{6 d} \]
command
integrate(cos(d*x+c)^7*(a+b*tan(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 \[ \text {Timed out} \]_______________________________________________________