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