\[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\left (-c^{4} x^{4}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{6 c^{8}}+\frac {b \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{18 c^{9} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{30 c^{9} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}+\frac {b \arctanh \left (\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-c^{2} x^{2}+1}}{3 c^{9} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}}{3 c^{9} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {-c^{4} x^{4}+1}}{2 c^{8}} \]
command
integrate(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (3 \, b c^{4} x^{4} + b c^{2} x^{2} + 28 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) + 15 \, {\left (a c^{6} x^{6} - a c^{4} x^{4} + 2 \, a c^{2} x^{2} + {\left (b c^{6} x^{6} - b c^{4} x^{4} + 2 \, b c^{2} x^{2} - 2 \, b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{90 \, {\left (c^{10} x^{2} - c^{8}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]