\[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (1+\left (-k^{2}-1\right ) x^{2}+k^{2} x^{4}\right )^{\frac {1}{3}}}{2 d^{\frac {1}{3}}-2 d^{\frac {1}{3}} k^{2} x^{2}+\left (1+\left (-k^{2}-1\right ) x^{2}+k^{2} x^{4}\right )^{\frac {1}{3}}}\right )}{2 d^{\frac {1}{3}}}+\frac {\ln \left (-d^{\frac {1}{3}}+d^{\frac {1}{3}} k^{2} x^{2}+\left (1+\left (-k^{2}-1\right ) x^{2}+k^{2} x^{4}\right )^{\frac {1}{3}}\right )}{2 d^{\frac {1}{3}}}-\frac {\ln \left (d^{\frac {2}{3}}-2 d^{\frac {2}{3}} k^{2} x^{2}+d^{\frac {2}{3}} k^{4} x^{4}+\left (d^{\frac {1}{3}}-d^{\frac {1}{3}} k^{2} x^{2}\right ) \left (1+\left (-k^{2}-1\right ) x^{2}+k^{2} x^{4}\right )^{\frac {1}{3}}+\left (1+\left (-k^{2}-1\right ) x^{2}+k^{2} x^{4}\right )^{\frac {2}{3}}\right )}{4 d^{\frac {1}{3}}} \]
command
Integrate[((1 - 2*k^2)*x + k^2*x^3)/(((1 - x^2)*(1 - k^2*x^2))^(1/3)*(-1 + d + (1 - 2*d*k^2)*x^2 + d*k^4*x^4)),x]
Mathematica 13.1 output
\[ \frac {\sqrt [3]{-1+x^2} \sqrt [3]{-1+k^2 x^2} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{\sqrt [3]{-1+x^2}-2 \sqrt [3]{d} \left (-1+k^2 x^2\right )^{2/3}}\right )+2 \log \left (\sqrt [3]{-1+x^2}+\sqrt [3]{d} \left (-1+k^2 x^2\right )^{2/3}\right )-\log \left (\left (-1+x^2\right )^{2/3}-\sqrt [3]{d} \sqrt [3]{-1+x^2} \left (-1+k^2 x^2\right )^{2/3}+d^{2/3} \left (-1+k^2 x^2\right )^{4/3}\right )\right )}{4 \sqrt [3]{d} \sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
Mathematica 12.3 output
\[ \int \frac {\left (1-2 k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (1-2 d k^2\right ) x^2+d k^4 x^4\right )} \, dx \]________________________________________________________________________________________