\[ \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx \]
Optimal antiderivative \[ \frac {\left (8 a^{2} A -3 A \,b^{2}+4 a b B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}} d}-\frac {\left (-i B +A \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {-i b +a}}\right )}{d \sqrt {-i b +a}}-\frac {\left (i B +A \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {i b +a}}\right )}{d \sqrt {i b +a}}+\frac {\left (3 A b -4 B a \right ) \cot \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )}}{4 a^{2} d}-\frac {A \left (\cot ^{2}\left (d x +c \right )\right ) \sqrt {a +b \tan \left (d x +c \right )}}{2 a d} \]
command
integrate(cot(d*x+c)^3*(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 \[ \text {Timed out} \]