\[ \int \frac {\csc ^3(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {7 \arctan \! \left (\frac {\sqrt {d \cos \left (b x +a \right )}}{\sqrt {d}}\right )}{4 b \,d^{\frac {5}{2}}}-\frac {7 \arctanh \! \left (\frac {\sqrt {d \cos \left (b x +a \right )}}{\sqrt {d}}\right )}{4 b \,d^{\frac {5}{2}}}+\frac {7}{6 b d \left (d \cos \! \left (b x +a \right )\right )^{\frac {3}{2}}}-\frac {\csc ^{2}\left (b x +a \right )}{2 b d \left (d \cos \! \left (b x +a \right )\right )^{\frac {3}{2}}} \]
command
integrate(csc(b*x+a)^3/(d*cos(b*x+a))^(5/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {could not integrate} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \frac {\frac {42 \, \arctan \left (-\frac {\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \frac {21 \, \log \left ({\left | -\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} \right |}\right )}{\sqrt {-d}} + \frac {6 \, \sqrt {-d}}{{\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} - d} - \frac {3 \, \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{d} + \frac {32 \, {\left (3 \, {\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}\right )}^{2} - d\right )}}{{\left (\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} - \sqrt {-d}\right )}^{3}}}{24 \, b d^{2}} \]