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