\[ \int \sqrt {\csc (a+b x)} \sec ^3(a+b x) \, dx \]
Optimal antiderivative \[ -\frac {3 \arctan \left (\sqrt {\csc }\left (b x +a \right )\right )}{4 b}+\frac {3 \arctanh \left (\sqrt {\csc }\left (b x +a \right )\right )}{4 b}+\frac {\sec ^{2}\left (b x +a \right )}{2 b \sqrt {\csc \left (b x +a \right )}} \]
command
integrate(csc(b*x+a)^(1/2)*sec(b*x+a)^3,x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\frac {4 \, \sqrt {\sin \left (b x + a\right )}}{\sin \left (b x + a\right )^{2} - 1} - 6 \, \arctan \left (\sqrt {\sin \left (b x + a\right )}\right ) - 3 \, \log \left (\sqrt {\sin \left (b x + a\right )} + 1\right ) + 3 \, \log \left ({\left | \sqrt {\sin \left (b x + a\right )} - 1 \right |}\right )}{8 \, b} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \sqrt {\csc \left (b x + a\right )} \sec \left (b x + a\right )^{3}\,{d x} \]________________________________________________________________________________________