\[ \int \frac {\sin (a+b x)}{(d \cos (a+b x))^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {2}{5 b d \left (d \cos \! \left (b x +a \right )\right )^{\frac {5}{2}}} \]
command
integrate(sin(b*x+a)/(d*cos(b*x+a))^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {2}{5 \, \sqrt {d \cos \left (b x + a\right )} b d^{3} \cos \left (b x + a\right )^{2}} \]