\[ \int \frac {1}{\left (a+b \csc ^2(c+d x)\right )^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctan \left (\frac {\cot \left (d x +c \right ) \sqrt {a}}{\sqrt {a +b +b \left (\cot ^{2}\left (d x +c \right )\right )}}\right )}{a^{\frac {7}{2}} d}+\frac {b \cot \left (d x +c \right )}{5 a \left (a +b \right ) d \left (a +b +b \left (\cot ^{2}\left (d x +c \right )\right )\right )^{\frac {5}{2}}}+\frac {b \left (9 a +5 b \right ) \cot \left (d x +c \right )}{15 a^{2} \left (a +b \right )^{2} d \left (a +b +b \left (\cot ^{2}\left (d x +c \right )\right )\right )^{\frac {3}{2}}}+\frac {b \left (33 a^{2}+40 a b +15 b^{2}\right ) \cot \left (d x +c \right )}{15 a^{3} \left (a +b \right )^{3} d \sqrt {a +b +b \left (\cot ^{2}\left (d x +c \right )\right )}} \]
command
integrate(1/(a+b*csc(d*x+c)^2)^(7/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \text {output too large to display} \]
Giac 1.7.0 via sagemath 9.3 output \[ \text {Exception raised: TypeError} \]_______________________________________________________