\[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \]
Optimal antiderivative \[ -\frac {b \arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a -b}}\right )}{4 \left (a -b \right )^{\frac {3}{2}} d}+\frac {b \arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a +b}}\right )}{4 \left (a +b \right )^{\frac {3}{2}} d}-\frac {2 \arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d \sqrt {a}}+\frac {\arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a -b}}\right )}{d \sqrt {a -b}}+\frac {\arctanh \left (\frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sqrt {a +b}}\right )}{d \sqrt {a +b}}+\frac {\sqrt {a +b \sec \left (d x +c \right )}}{4 \left (a +b \right ) d \left (1-\sec \left (d x +c \right )\right )}+\frac {\sqrt {a +b \sec \left (d x +c \right )}}{4 \left (a -b \right ) d \left (1+\sec \left (d x +c \right )\right )} \]
command
integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {\frac {{\left (4 \, a - 5 \, b\right )} \sqrt {a - b} \log \left ({\left | {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} {\left (a - b\right )} - \sqrt {a - b} a \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {16 \, \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2 \, {\left (4 \, a + 5 \, b\right )} \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}}{\sqrt {-a - b}}\right )}{{\left (a + b\right )} \sqrt {-a - b}} + \frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}}{a - b} - \frac {2 \, {\left ({\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} a - {\left (a + b\right )} \sqrt {a - b}\right )}}{{\left ({\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{2} - a - b\right )} {\left (a + b\right )}}}{8 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: TypeError} \]________________________________________________________________________________________