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