\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx \]
Optimal antiderivative \[ -\frac {i \arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d +c}}\right ) \sqrt {-i d +c}}{\left (-i b +a \right )^{2} f}+\frac {i \arctanh \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d +c}}\right ) \sqrt {i d +c}}{\left (i b +a \right )^{2} f}-\frac {\left (-3 a^{2} d +4 a b c +b^{2} d \right ) \arctanh \left (\frac {\sqrt {b}\, \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-a d +b c}}\right ) \sqrt {b}}{\left (a^{2}+b^{2}\right )^{2} f \sqrt {-a d +b c}}-\frac {b \sqrt {c +d \tan \left (f x +e \right )}}{\left (a^{2}+b^{2}\right ) f \left (a +b \tan \left (f x +e \right )\right )} \]
command
integrate((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e))^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]