\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\arctanh \left (\frac {\sqrt {3-2 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2+3 \tan \left (d x +c \right )}}\right )}{d \sqrt {3-2 i}}+\frac {\arctanh \left (\frac {\sqrt {3+2 i}\, \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2+3 \tan \left (d x +c \right )}}\right )}{d \sqrt {3+2 i}} \]
command
integrate(1/tan(d*x+c)^(1/2)/(2+3*tan(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {\sqrt {3} {\left (\left (3 i + 2\right ) \, \sqrt {6 \, \sqrt {13} - 18} {\left (-\frac {2 i}{\sqrt {13} - 3} + 1\right )} \log \left (\left (120 i + 40\right ) \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + \left (432 i + 144\right ) \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + 80 \, \sqrt {13} \sqrt {15 \, \sqrt {13} + 54} - 800 \, \sqrt {13} - \left (16 i - 288\right ) \, \sqrt {15 \, \sqrt {13} + 54} - 2880\right ) - \left (3 i + 2\right ) \, \sqrt {6 \, \sqrt {13} - 18} {\left (-\frac {2 i}{\sqrt {13} - 3} + 1\right )} \log \left (\left (120 i + 40\right ) \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + \left (432 i + 144\right ) \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 80 \, \sqrt {13} \sqrt {15 \, \sqrt {13} + 54} - 800 \, \sqrt {13} + \left (16 i - 288\right ) \, \sqrt {15 \, \sqrt {13} + 54} - 2880\right ) + \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )} \log \left (8 \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 24 \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + 8 \, \sqrt {13} \sqrt {6 \, \sqrt {13} - 18} - \left (48 i + 16\right ) \, \sqrt {13} + \left (16 i - 24\right ) \, \sqrt {6 \, \sqrt {13} - 18} + 144 i + 48\right ) - \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )} \log \left (8 \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 24 \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 8 \, \sqrt {13} \sqrt {6 \, \sqrt {13} - 18} - \left (48 i + 16\right ) \, \sqrt {13} - \left (16 i - 24\right ) \, \sqrt {6 \, \sqrt {13} - 18} + 144 i + 48\right )\right )}}{156 \, d} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {3 \, \tan \left (d x + c\right ) + 2} \sqrt {\tan \left (d x + c\right )}}\,{d x} \]________________________________________________________________________________________