\[ \int \frac {x^3 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx \]
Optimal antiderivative \[ \frac {b x \arctan \left (\frac {\sqrt {-c^{4} x^{4}+1}}{\sqrt {c^{2} x^{2}-1}}\right )}{2 c^{3} \sqrt {c^{2} x^{2}}}-\frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {-c^{4} x^{4}+1}}{2 c^{4}}-\frac {b x \sqrt {-c^{4} x^{4}+1}}{2 c^{3} \sqrt {c^{2} x^{2}}\, \sqrt {c^{2} x^{2}-1}} \]
command
integrate(x^3*(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
\[ -\frac {\sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} b - {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) + \sqrt {-c^{4} x^{4} + 1} {\left (a c^{2} x^{2} + {\left (b c^{2} x^{2} - b\right )} \operatorname {arccsc}\left (c x\right ) - a\right )}}{2 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]