\[ \int \frac {\sec ^3(a+b x)}{\sqrt {\csc (a+b x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\sqrt {\csc }\left (b x +a \right )\right )}{4 b}+\frac {\arctanh \left (\sqrt {\csc }\left (b x +a \right )\right )}{4 b}+\frac {\sec ^{2}\left (b x +a \right )}{2 b \csc \left (b x +a \right )^{\frac {3}{2}}} \]
command
integrate(sec(b*x+a)^3/csc(b*x+a)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {4 \, \sin \left (b x + a\right )^{\frac {3}{2}}}{\sin \left (b x + a\right )^{2} - 1} + 2 \, \arctan \left (\sqrt {\sin \left (b x + a\right )}\right ) - \log \left (\sqrt {\sin \left (b x + a\right )} + 1\right ) + \log \left ({\left | \sqrt {\sin \left (b x + a\right )} - 1 \right |}\right )}{8 \, b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {\sec \left (b x + a\right )^{3}}{\sqrt {\csc \left (b x + a\right )}}\,{d x} \]________________________________________________________________________________________