\[ \int (b \sec (c+d x)+a \sin (c+d x))^n (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {\left (b \sec \left (d x +c \right )+a \sin \left (d x +c \right )\right )^{1+n}}{d \left (1+n \right )} \]
command
integrate((b*sec(d*x+c)+a*sin(d*x+c))^n*(a*cos(d*x+c)+b*sec(d*x+c)*tan(d*x+c)),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\left (-\frac {b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1}\right )^{n + 1}}{d {\left (n + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int {\left (b \sec \left (d x + c\right ) \tan \left (d x + c\right ) + a \cos \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a \sin \left (d x + c\right )\right )}^{n}\,{d x} \]________________________________________________________________________________________