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