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