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