Integrand size = 59, antiderivative size = 241 \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}{2 \sqrt [3]{d}-2 \sqrt [3]{d} x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}}\right )}{2 \sqrt [3]{d}}+\frac {\log \left (-\sqrt [3]{d}+\sqrt [3]{d} x^2+\sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}\right )}{2 \sqrt [3]{d}}-\frac {\log \left (d^{2/3}-2 d^{2/3} x^2+d^{2/3} x^4+\left (\sqrt [3]{d}-\sqrt [3]{d} x^2\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}+\left (1+\left (-1-k^2\right ) x^2+k^2 x^4\right )^{2/3}\right )}{4 \sqrt [3]{d}} \]
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\[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-2+k^2+k^2 x^2\right )}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx \\ & = \int \frac {x \left (-2+k^2+k^2 x^2\right )}{\left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right ) \sqrt [3]{1+\left (-1-k^2\right ) x^2+k^2 x^4}} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {-2+k^2+k^2 x}{\left (-1+d+\left (-2 d+k^2\right ) x+d x^2\right ) \sqrt [3]{1+\left (-1-k^2\right ) x+k^2 x^2}} \, dx,x,x^2\right ) \\ \end{align*}
Time = 15.23 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\frac {\sqrt [3]{-1+x^2} \sqrt [3]{-1+k^2 x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+k^2 x^2}}{-2 \sqrt [3]{d} \left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+k^2 x^2}}\right )+2 \log \left (\sqrt [3]{d} \left (-1+x^2\right )^{2/3}+\sqrt [3]{-1+k^2 x^2}\right )-\log \left (d^{2/3} \left (-1+x^2\right )^{4/3}-\sqrt [3]{d} \left (-1+x^2\right )^{2/3} \sqrt [3]{-1+k^2 x^2}+\left (-1+k^2 x^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{d} \sqrt [3]{\left (-1+x^2\right ) \left (-1+k^2 x^2\right )}} \]
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\[\int \frac {\left (k^{2}-2\right ) x +k^{2} x^{3}}{{\left (\left (-x^{2}+1\right ) \left (-k^{2} x^{2}+1\right )\right )}^{\frac {1}{3}} \left (-1+d +\left (k^{2}-2 d \right ) x^{2}+d \,x^{4}\right )}d x\]
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Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int { \frac {k^{2} x^{3} + {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int { \frac {k^{2} x^{3} + {\left (k^{2} - 2\right )} x}{{\left (d x^{4} + {\left (k^{2} - 2 \, d\right )} x^{2} + d - 1\right )} \left ({\left (k^{2} x^{2} - 1\right )} {\left (x^{2} - 1\right )}\right )^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (-2+k^2\right ) x+k^2 x^3}{\sqrt [3]{\left (1-x^2\right ) \left (1-k^2 x^2\right )} \left (-1+d+\left (-2 d+k^2\right ) x^2+d x^4\right )} \, dx=\int \frac {x\,\left (k^2-2\right )+k^2\,x^3}{{\left (\left (x^2-1\right )\,\left (k^2\,x^2-1\right )\right )}^{1/3}\,\left (d\,x^4+\left (k^2-2\,d\right )\,x^2+d-1\right )} \,d x \]
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