\[ \int \frac {1}{\sqrt {a+a \tan ^2(c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\tan \left (d x +c \right )}{d \sqrt {a \left (\sec ^{2}\left (d x +c \right )\right )}} \]
command
integrate(1/(a+a*tan(d*x+c)^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {2}{\sqrt {a} d {\left (\frac {1}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {1}{\sqrt {a \tan \left (d x + c\right )^{2} + a}}\,{d x} \]________________________________________________________________________________________