Optimal. Leaf size=158 \[ -\frac {\sqrt [3]{1-x^3}}{3 x^3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {457, 105, 162,
59, 632, 210, 31, 631} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\sqrt [3]{1-x^3}}{3 x^3}-\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac {\log (x)}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 59
Rule 105
Rule 162
Rule 210
Rule 457
Rule 631
Rule 632
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x)^{2/3} x^2 (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1-x^3}}{3 x^3}-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {1}{3}-\frac {2 x}{3}}{(1-x)^{2/3} x (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1-x^3}}{3 x^3}-\frac {1}{9} \text {Subst}\left (\int \frac {1}{(1-x)^{2/3} x} \, dx,x,x^3\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x)^{2/3} (1+x)} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{1-x^3}}{3 x^3}+\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1-x^3}\right )+\frac {1}{6} \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\sqrt [3]{1-x^3}}{3 x^3}+\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1-x^3}\right )+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{2^{2/3}}\\ &=-\frac {\sqrt [3]{1-x^3}}{3 x^3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log (x)}{6}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {1}{6} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac {\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 202, normalized size = 1.28 \begin {gather*} \frac {1}{36} \left (-\frac {12 \sqrt [3]{1-x^3}}{x^3}+4 \sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-6 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt {3}}\right )-4 \log \left (-1+\sqrt [3]{1-x^3}\right )+6 \sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x^3}\right )+2 \log \left (1+\sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )-3 \sqrt [3]{2} \log \left (2+2^{2/3} \sqrt [3]{1-x^3}+\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (-x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+1\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.99, size = 195, normalized size = 1.23 \begin {gather*} -\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{3} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} {\left (4^{\frac {2}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \sqrt {3}\right )}\right ) + 3 \cdot 4^{\frac {2}{3}} x^{3} \log \left (4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2 \cdot 4^{\frac {1}{3}}\right ) - 6 \cdot 4^{\frac {2}{3}} x^{3} \log \left (-4^{\frac {2}{3}} + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right ) - 8 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 4 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + 8 \, x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1\right ) + 24 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{72 \, x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{4} \left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.90, size = 163, normalized size = 1.03 \begin {gather*} -\frac {1}{6} \, \sqrt {3} 2^{\frac {1}{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}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{12} \cdot 2^{\frac {1}{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}{6} \cdot 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} \right |}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x^{3}} + \frac {1}{18} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac {2}{3}} + {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{9} \, \log \left ({\left | {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.07, size = 368, normalized size = 2.33 \begin {gather*} \frac {2^{1/3}\,\ln \left (\frac {10\,{\left (1-x^3\right )}^{1/3}}{9}-\frac {2^{1/3}\,\left (\frac {2^{2/3}\,\left (243\,2^{1/3}+27\,{\left (1-x^3\right )}^{1/3}\right )}{36}-\frac {25}{3}\right )}{6}\right )}{6}-\frac {{\left (1-x^3\right )}^{1/3}}{3\,x^3}-\frac {\ln \left (\frac {31\,{\left (1-x^3\right )}^{1/3}}{243}-\frac {31}{243}\right )}{9}-\ln \left (\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )\,\left ({\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\,\left (27\,{\left (1-x^3\right )}^{1/3}+81-\sqrt {3}\,81{}\mathrm {i}\right )-\frac {25}{3}\right )+\frac {10\,{\left (1-x^3\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\ln \left (\frac {10\,{\left (1-x^3\right )}^{1/3}}{9}-\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )\,\left ({\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\,\left (27\,{\left (1-x^3\right )}^{1/3}+81+\sqrt {3}\,81{}\mathrm {i}\right )-\frac {25}{3}\right )\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )+\frac {2^{1/3}\,\ln \left (\frac {10\,{\left (1-x^3\right )}^{1/3}}{9}-\frac {2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {2^{2/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {243\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+27\,{\left (1-x^3\right )}^{1/3}\right )}{144}-\frac {25}{3}\right )}{12}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{12}-\frac {2^{1/3}\,\ln \left (\frac {10\,{\left (1-x^3\right )}^{1/3}}{9}-\frac {2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (\frac {2^{2/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (\frac {243\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-27\,{\left (1-x^3\right )}^{1/3}\right )}{144}+\frac {25}{3}\right )}{12}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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