Optimal. Leaf size=135 \[ -\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \log \left (x+\sqrt [3]{1-x^3}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {494, 245, 384}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}+\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 384
Rule 494
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\text {Subst}\left (\int \frac {x^3}{\left (1+x^3\right ) \left (1+2 x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\text {Subst}\left (\int \frac {1}{1+x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )-\text {Subst}\left (\int \frac {1}{1+2 x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {2-x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {2-\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {1}{6} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}\\ &=-\frac {1}{6} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{\sqrt [3]{2}}-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+\frac {2 x}{\sqrt [3]{1-x^3}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {-1+\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {1}{6} \log \left (1+\frac {x^2}{\left (1-x^3\right )^{2/3}}-\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {1}{3} \log \left (1+\frac {x}{\sqrt [3]{1-x^3}}\right )+\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}-\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 205, normalized size = 1.52 \begin {gather*} \frac {1}{12} \left (-4 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3}}\right )+2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )+4 \log \left (x+\sqrt [3]{1-x^3}\right )-2\ 2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )-2 \log \left (x^2-x \sqrt [3]{1-x^3}+\left (1-x^3\right )^{2/3}\right )+2^{2/3} \log \left (-2 x^2+2^{2/3} x \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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}\, dx\]
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 [C] Result contains complex when optimal does not.
time = 7.21, size = 452, normalized size = 3.35 \begin {gather*} \frac {1}{12} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} \log \left (-\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} - 6 \cdot 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, x - 24 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{24} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} - 8 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \frac {1}{24} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \log \left (\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} + 32 \, x + 24 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{24 \, x}\right ) - \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \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 {x^{3}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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