\[ \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx \]
Optimal antiderivative \[ \frac {\left (8 A \,a^{2} b -A \,b^{3}+16 a^{3} B +2 B a \,b^{2}\right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{8 a^{\frac {5}{2}} d}-\frac {\left (i A +B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {-i b +a}}\right ) \sqrt {-i b +a}}{d}+\frac {\left (i A -B \right ) \arctanh \left (\frac {\sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {i b +a}}\right ) \sqrt {i b +a}}{d}+\frac {\left (8 a^{2} A +A \,b^{2}-2 a b B \right ) \cot \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )}}{8 a^{2} d}-\frac {\left (A b +6 B a \right ) \left (\cot ^{2}\left (d x +c \right )\right ) \sqrt {a +b \tan \left (d x +c \right )}}{12 a d}-\frac {A \left (\cot ^{3}\left (d x +c \right )\right ) \sqrt {a +b \tan \left (d x +c \right )}}{3 d} \]
command
integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),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 \[ \text {Timed out} \]