\[ \int \frac {\cot ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}} d}+\frac {\left (i A +B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {-i b +a}}\right )}{d \sqrt {-i b +a}}-\frac {\left (i A -B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {i b +a}}\right )}{d \sqrt {i b +a}}-\frac {A \cot \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )}}{a d} \]
command
integrate(cot(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),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 \[ \text {Timed out} \]