\[ \int \sec ^2(2 (a+b x)) \sqrt {c \tan (a+b x) \tan (2 (a+b x))} \, dx \]
Optimal antiderivative \[ -\frac {c \tan \left (2 b x +2 a \right )}{3 b \sqrt {-c +c \sec \left (2 b x +2 a \right )}}+\frac {\sqrt {-c +c \sec \left (2 b x +2 a \right )}\, \tan \left (2 b x +2 a \right )}{3 b} \]
command
integrate(sec(2*b*x+2*a)^2*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Timed out} \]_______________________________________________________