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