\[ \int \frac {1}{\sqrt {-3-2 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {2-3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3-2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2-3 i}}+\frac {\arctan \left (\frac {\sqrt {2+3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3-2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2+3 i}} \]
command
integrate(1/(-3-2*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
\[ -\frac {\left (1904 i + 1536\right ) \, \sqrt {2} \log \left (3 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} + \left (24 i + 18\right ) \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 27\right )}{23377 \, d} - \frac {\left (1904 i - 1536\right ) \, \sqrt {2} \log \left (3 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} - \left (24 i - 18\right ) \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 27\right )}{23377 \, d} - \frac {\left (12720 i + 27456\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 2 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (4 i - 3\right ) \, \sqrt {13} - 8 i + 6}{2 \, {\left (\sqrt {13} \sqrt {\sqrt {13} + 2} + \left (3 i + 2\right ) \, \sqrt {\sqrt {13} + 2}\right )}}\right )}{23377 \, d \sqrt {\sqrt {13} + 2} {\left (\frac {3 i}{\sqrt {13} + 2} + 1\right )}} - \frac {\left (12720 i - 27456\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 2 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (4 i + 3\right ) \, \sqrt {13} + 8 i + 6}{2 \, {\left (\sqrt {13} \sqrt {\sqrt {13} + 2} - \left (3 i - 2\right ) \, \sqrt {\sqrt {13} + 2}\right )}}\right )}{23377 \, d \sqrt {\sqrt {13} + 2} {\left (-\frac {3 i}{\sqrt {13} + 2} + 1\right )}} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {-2 \, \tan \left (d x + c\right ) - 3} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]________________________________________________________________________________________