Optimal. Leaf size=100 \[ \frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]
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Rubi [A] time = 0.07, antiderivative size = 100, 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 = {444, 51, 55, 617, 204, 31} \[ \frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 444
Rule 617
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x)^{4/3} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=\frac {1}{2 \sqrt [3]{1-x^3}}+\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{12 \sqrt [3]{2}}+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.34 \[ \frac {\, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {1}{2} \left (1-x^3\right )\right )}{2 \sqrt [3]{1-x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 125, normalized size = 1.25 \[ \frac {2 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{3} - 1\right )} \arctan \left (\frac {1}{6} \cdot 2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} + 2 \, \sqrt {6} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + 2 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{24 \, {\left (x^{3} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 98, normalized size = 0.98 \[ \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.87, size = 667, normalized size = 6.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 97, normalized size = 0.97 \[ \frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.90, size = 117, normalized size = 1.17 \[ \frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}}{4}\right )}{12}+\frac {1}{2\,{\left (1-x^3\right )}^{1/3}}+\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{24}-\frac {2^{2/3}\,\ln \left (\frac {{\left (1-x^3\right )}^{1/3}}{4}-\frac {2^{1/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{16}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{24} \]
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 \[ \int \frac {x^{2}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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