\[ \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (a +b \sin \! \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b^{3} d}+\frac {2 a^{2}-2 b^{2}}{b^{3} d \sqrt {a +b \sin \! \left (d x +c \right )}}+\frac {4 a \sqrt {a +b \sin \! \left (d x +c \right )}}{b^{3} d} \]
command
integrate(cos(d*x+c)^3/(a+b*sin(d*x+c))^(3/2),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
\[ \frac {2 \, {\left (\frac {3 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {b \sin \left (d x + c\right ) + a} b^{3}} - \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b \sin \left (d x + c\right ) + a} a b^{6}}{b^{9}}\right )}}{3 \, d} \]