\[ \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2}{3 b d \left (a +b \sin \! \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate(cos(d*x+c)/(a+b*sin(d*x+c))^(5/2),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
\[ -\frac {2}{3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b d} \]