\[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {3-2 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-2-3 \tan \left (d x +c \right )}}\right )}{d \sqrt {3-2 i}}+\frac {\arctan \left (\frac {\sqrt {3+2 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-2-3 \tan \left (d x +c \right )}}\right )}{d \sqrt {3+2 i}} \]
command
integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} - \left (6 i - 2\right ) \, \sqrt {13} - 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} + \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (\frac {2 i}{\sqrt {13} + 3} + 1\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + \left (6 i + 2\right ) \, \sqrt {13} + 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} - \left (2 i - 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {-3 \, \tan \left (d x + c\right ) - 2} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]________________________________________________________________________________________