\[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\frac {2 a^{2}}{3}-\frac {2 b^{2}}{3}}{b^{3} d \left (a +b \sin \! \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {4 a}{b^{3} d \sqrt {a +b \sin \! \left (d x +c \right )}}-\frac {2 \sqrt {a +b \sin \! \left (d x +c \right )}}{b^{3} d} \]
command
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {Timed out} \]
Giac 1.7.0 via sagemath 9.3 output
\[ -\frac {2 \, {\left (\frac {3 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b^{3}} + \frac {6 \, {\left (b \sin \left (d x + c\right ) + a\right )} a - a^{2} + b^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b^{3}}\right )}}{3 \, d} \]