\[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {\frac {1}{2}+\frac {\sin \left (2 b x +2 a \right )}{2}}\, \EllipticF \left (\cos \left (a +\frac {\pi }{4}+b x \right ), \sqrt {2}\right )}{6 \sin \left (a +\frac {\pi }{4}+b x \right ) b}-\frac {\cos ^{2}\left (b x +a \right )}{3 b \sin \left (2 b x +2 a \right )^{\frac {3}{2}}} \]
command
integrate(cos(b*x+a)^2/sin(2*b*x+2*a)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {\sqrt {2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ), -1\right ) + \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{2} - 1\right )} {\rm ellipticF}\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ), -1\right ) - \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{12 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {\cos \left (b x + a\right )^{2}}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} - 1\right )} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}}, x\right ) \]