\[ \int \frac {\cos ^2(a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx \]
Optimal antiderivative \[ \frac {3 \sqrt {\frac {1}{2}+\frac {\sin \left (2 b x +2 a \right )}{2}}\, \EllipticE \! \left (\cos \! \left (a +\frac {\pi }{4}+b x \right ), \sqrt {2}\right )}{10 \sin \! \left (a +\frac {\pi }{4}+b x \right ) b}-\frac {\cos ^{2}\left (b x +a \right )}{5 b \sin \! \left (2 b x +2 a \right )^{\frac {5}{2}}}-\frac {3 \cos \! \left (2 b x +2 a \right )}{10 b \sqrt {\sin \! \left (2 b x +2 a \right )}} \]
command
int(cos(b*x+a)^2/sin(2*b*x+2*a)^(7/2),x)
Maple 2022.1 output
\[\int \frac {\cos ^{2}\left (x b +a \right )}{\sin \left (2 x b +2 a \right )^{\frac {7}{2}}}\, dx\]
Maple 2021.1 output
\[ \frac {\sqrt {2}\, \left (-\frac {8 \sqrt {2}}{5 \sin \left (2 b x +2 a \right )^{\frac {5}{2}}}+\frac {4 \sqrt {2}\, \left (6 \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \left (\sin ^{2}\left (2 b x +2 a \right )\right ) \EllipticE \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1+\sin \left (2 b x +2 a \right )}\, \sqrt {-2 \sin \left (2 b x +2 a \right )+2}\, \sqrt {-\sin \left (2 b x +2 a \right )}\, \left (\sin ^{2}\left (2 b x +2 a \right )\right ) \EllipticF \left (\sqrt {1+\sin \left (2 b x +2 a \right )}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{4}\left (2 b x +2 a \right )\right )-4 \left (\sin ^{2}\left (2 b x +2 a \right )\right )-2\right )}{5 \sin \left (2 b x +2 a \right )^{\frac {5}{2}} \cos \left (2 b x +2 a \right )}\right )}{32 b} \]