Optimal. Leaf size=515 \[ -\frac {3 x}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}(x)}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.17, antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {483, 235, 304, 219, 1879, 393} \[ -\frac {3 x}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}-\frac {3 \tanh ^{-1}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{2\ 2^{2/3}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{2 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}(x)}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 393
Rule 483
Rule 1879
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx &=-\left (3 \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx\right )+\int \frac {1}{\sqrt [3]{1-x^2}} \, dx\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac {3 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}-\frac {\left (3 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{2 x}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac {3 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}+\frac {\left (3 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{2 x}-\frac {\left (3 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{x}\\ &=-\frac {3 x}{1-\sqrt {3}-\sqrt [3]{1-x^2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{2\ 2^{2/3}}+\frac {\tanh ^{-1}(x)}{2\ 2^{2/3}}-\frac {3 \tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{2\ 2^{2/3}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{2 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 28, normalized size = 0.05 \[ \frac {1}{9} x^3 F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};x^2,-\frac {x^2}{3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 3.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-x^{2} + 1\right )}^{\frac {2}{3}} x^{2}}{x^{4} + 2 \, x^{2} - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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