\[ \int \frac {x^{11} \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\left (-c^{4} x^{4}+1\right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{3 c^{12}}-\frac {\left (-c^{4} x^{4}+1\right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )}{10 c^{12}}+\frac {7 b \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \sqrt {-c^{2} x^{2}+1}}{90 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {13 b \left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \sqrt {-c^{2} x^{2}+1}}{150 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}+\frac {3 b \left (c^{2} x^{2}+1\right )^{\frac {7}{2}} \sqrt {-c^{2} x^{2}+1}}{70 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {b \left (c^{2} x^{2}+1\right )^{\frac {9}{2}} \sqrt {-c^{2} x^{2}+1}}{90 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}+\frac {4 b \arctanh \left (\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-c^{2} x^{2}+1}}{15 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {4 b \sqrt {-c^{2} x^{2}+1}\, \sqrt {c^{2} x^{2}+1}}{15 c^{13} x \sqrt {1-\frac {1}{c^{2} x^{2}}}}-\frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {-c^{4} x^{4}+1}}{2 c^{12}} \]
command
integrate(x^11*(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
\[ -\frac {{\left (35 \, b c^{8} x^{8} + 5 \, b c^{6} x^{6} + 78 \, b c^{4} x^{4} + 36 \, b c^{2} x^{2} + 768 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} - 840 \, {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1}}{\sqrt {c^{2} x^{2} - 1}}\right ) + 105 \, {\left (3 \, a c^{10} x^{10} - 3 \, a c^{8} x^{8} + 4 \, a c^{6} x^{6} - 4 \, a c^{4} x^{4} + 8 \, a c^{2} x^{2} + {\left (3 \, b c^{10} x^{10} - 3 \, b c^{8} x^{8} + 4 \, b c^{6} x^{6} - 4 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} - 8 \, b\right )} \operatorname {arccsc}\left (c x\right ) - 8 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{3150 \, {\left (c^{14} x^{2} - c^{12}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]