\[ \int \frac {A+C \cot ^2(c+d x)}{\sqrt {b \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {\left (A -C \right ) \arctan \left (1-\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {2}}{2 d \sqrt {b}}+\frac {\left (A -C \right ) \arctan \left (1+\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}}{\sqrt {b}}\right ) \sqrt {2}}{2 d \sqrt {b}}-\frac {\left (A -C \right ) \ln \left (\sqrt {b}-\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\sqrt {b}\, \tan \left (d x +c \right )\right ) \sqrt {2}}{4 d \sqrt {b}}+\frac {\left (A -C \right ) \ln \left (\sqrt {b}+\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\sqrt {b}\, \tan \left (d x +c \right )\right ) \sqrt {2}}{4 d \sqrt {b}}-\frac {2 b C}{3 d \left (b \tan \left (d x +c \right )\right )^{\frac {3}{2}}} \]
command
integrate((A+C*cot(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{2 \, b d} + \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {\sqrt {2} {\left (A \sqrt {{\left | b \right |}} - C \sqrt {{\left | b \right |}}\right )} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{4 \, b d} - \frac {2 \, C}{3 \, \sqrt {b \tan \left (d x + c\right )} d \tan \left (d x + c\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {C \cot \left (d x + c\right )^{2} + A}{\sqrt {b \tan \left (d x + c\right )}}\,{d x} \]________________________________________________________________________________________