\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {2-3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3+2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2-3 i}}+\frac {\arctanh \left (\frac {\sqrt {2+3 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {-3+2 \tan \left (d x +c \right )}}\right )}{d \sqrt {2+3 i}} \]
command
integrate(1/tan(d*x+c)^(1/2)/(-3+2*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {2} {\left (\sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} - \left (1647 i - 6954\right ) \, \sqrt {13} - \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} - 4266 i + 18012\right ) - \sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} - \left (1647 i - 6954\right ) \, \sqrt {13} + \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} - 4266 i + 18012\right ) - \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} + \left (450 i - 225\right ) \, \sqrt {13} - \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} - 540 i + 270\right ) + \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} + \left (450 i - 225\right ) \, \sqrt {13} + \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} - 540 i + 270\right )\right )}}{52 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {2 \, \tan \left (d x + c\right ) - 3} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]________________________________________________________________________________________