\[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {8 a \left (a^{2}-b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b^{5} d}+\frac {4 \left (3 a^{2}-b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 b^{5} d}-\frac {8 a \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 b^{5} d}+\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 b^{5} d}+\frac {2 \left (a^{2}-b^{2}\right )^{2} \sqrt {a +b \sin \left (d x +c \right )}}{b^{5} d} \]
command
integrate(cos(d*x+c)^5/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2 \, {\left (35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4} - 126 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} b^{2} + 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a b^{2} - 630 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2} b^{2} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} b^{4}\right )}}{315 \, b^{5} d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\cos \left (d x + c\right )^{5}}{\sqrt {b \sin \left (d x + c\right ) + a}}\,{d x} \]________________________________________________________________________________________