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