\[ \int \frac {\csc (a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx \]
Optimal antiderivative \[ -\frac {2 \cos \left (b x +a \right )}{7 b \sin \left (2 b x +2 a \right )^{\frac {7}{2}}}+\frac {12 \sin \left (b x +a \right )}{35 b \sin \left (2 b x +2 a \right )^{\frac {5}{2}}}-\frac {16 \cos \left (b x +a \right )}{35 b \sin \left (2 b x +2 a \right )^{\frac {3}{2}}}+\frac {32 \sin \left (b x +a \right )}{35 b \sqrt {\sin \left (2 b x +2 a \right )}} \]
command
int(csc(b*x+a)/sin(2*b*x+2*a)^(7/2),x)
Maple 2022.1 output
\[\frac {\sqrt {-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right ) \left (3 \left (\tan ^{8}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+40 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-26 \left (\tan ^{6}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )+26 \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )\right )-3\right )}{1344 b \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3} \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan ^{2}\left (\frac {a}{2}+\frac {x b}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {a}{2}+\frac {x b}{2}\right )-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}}\]
Maple 2021.1 output
\[ \int \frac {\csc \left (b x +a \right )}{\sin \left (2 b x +2 a \right )^{\frac {7}{2}}}\, dx \]________________________________________________________________________________________