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