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