Optimal. Leaf size=53 \[ \frac {1}{3} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )-\frac {2}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1960, 288, 204} \[ \frac {1}{3} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )-\frac {2}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 288
Rule 1960
Rubi steps
\begin {align*} \int x^2 \sqrt {\frac {1-x^3}{1+x^3}} \, dx &=-\left (\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\right )\\ &=\frac {1}{3} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{3} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {2}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 86, normalized size = 1.62 \[ \frac {\sqrt {\frac {1-x^3}{x^3+1}} \sqrt {x^3+1} \left (\sqrt {x^3+1} \left (x^3-1\right )+2 \sqrt {1-x^3} \sin ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {2}}\right )\right )}{3 \left (x^3-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 55, normalized size = 1.04 \[ \frac {1}{3} \, {\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - \frac {2}{3} \, \arctan \left (\frac {{\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 22, normalized size = 0.42 \[ \frac {1}{3} \, {\left (\sqrt {-x^{6} + 1} + \arcsin \left (x^{3}\right )\right )} \mathrm {sgn}\left (x^{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 68, normalized size = 1.28 \[ -\frac {\sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, \sqrt {-\left (x^{3}+1\right ) \left (x^{3}-1\right )}\, \arcsin \left (x^{3}\right )}{3 \left (x^{3}-1\right )}+\frac {\left (x^{3}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}}{3} \]
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 \[ \int x^{2} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.67, size = 56, normalized size = 1.06 \[ -\frac {2\,\mathrm {atan}\left (\sqrt {-\frac {x^3-1}{x^3+1}}\right )}{3}-\frac {2\,\sqrt {-\frac {x^3-1}{x^3+1}}}{\frac {3\,\left (x^3-1\right )}{x^3+1}-3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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