Optimal. Leaf size=100 \[ -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(505\) vs. \(2(100)=200\).
time = 0.41, antiderivative size = 505, normalized size of antiderivative = 5.05, number of steps
used = 12, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283,
245, 399, 384} \begin {gather*} -\frac {\left (3+\sqrt {5}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\left (3-\sqrt {5}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {15}}+\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \text {ArcTan}\left (\frac {1-\frac {2^{2/3} \sqrt [3]{\sqrt {5}-1} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \text {ArcTan}\left (\frac {\frac {2^{2/3} \sqrt [3]{1+\sqrt {5}} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {15}}-\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (2 x^3-\sqrt {5}-1\right )}{6 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (2 x^3+\sqrt {5}-1\right )}{6 \sqrt {5}}-\frac {\sqrt [3]{\frac {1}{2} \left (7+3 \sqrt {5}\right )} \log \left (-\sqrt [3]{x^3+1}-\sqrt [3]{\frac {1}{2} \left (\sqrt {5}-1\right )} x\right )}{2 \sqrt {5}}+\frac {\sqrt [3]{\frac {1}{2} \left (7-3 \sqrt {5}\right )} \log \left (\sqrt [3]{\frac {1}{2} \left (1+\sqrt {5}\right )} x-\sqrt [3]{x^3+1}\right )}{2 \sqrt {5}}+\frac {1}{20} \left (5+3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{20} \left (5-3 \sqrt {5}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 283
Rule 384
Rule 399
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-1-x^3+x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6}\right ) \, dx\\ &=\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^3\right ) \left (1+x^3\right )^{2/3}}{-1-x^3+x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (-1+\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3}+\frac {\left (-1-\frac {3}{\sqrt {5}}\right ) \left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1-\sqrt {5}+2 x^3} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-1+\sqrt {5}+2 x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (5+3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {5}}\right )}{5 \left (1-\sqrt {5}\right )}+\frac {\left (5-3 \sqrt {5}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {5}}\right )}{5 \left (1+\sqrt {5}\right )}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 100, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \text {RootSum}\left [-1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+2 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 144.16, size = 9494, normalized size = 94.94
method | result | size |
trager | \(\text {Expression too large to display}\) | \(9494\) |
risch | \(\text {Expression too large to display}\) | \(11358\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (x^3-1\right )\,{\left (x^3+1\right )}^{2/3}}{x^3\,\left (-x^6+x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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