Optimal. Leaf size=103 \[ -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(103)=206\).
time = 0.61, antiderivative size = 327, normalized size of antiderivative = 3.17, number of steps
used = 10, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6860, 283,
245, 1532, 384} \begin {gather*} \frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (4 x^3-\sqrt {17}+1\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (4 x^3+\sqrt {17}+1\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (5 \sqrt {17}-19\right )} \log \left (\frac {\sqrt [3]{5-\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\frac {\sqrt [3]{5+\sqrt {17}} x}{2^{2/3}}-\sqrt [3]{x^3+1}\right )}{2 \sqrt {17}}-\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 1532
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (-2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^3}-\frac {2 x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {x^3 \left (1+x^3\right )^{2/3}}{-2+x^3+2 x^6} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \frac {-2-x^3}{\sqrt [3]{1+x^3} \left (-2+x^3+2 x^6\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\int \left (\frac {-1-\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )}+\frac {-1+\frac {7}{\sqrt {17}}}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )}\right ) \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1+\sqrt {17}+4 x^3\right )} \, dx-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \int \frac {1}{\sqrt [3]{1+x^3} \left (1-\sqrt {17}+4 x^3\right )} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{17} \left (-17+7 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{17} \left (17+7 \sqrt {17}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{71+17 \sqrt {17}} \text {Subst}\left (\int \frac {1}{-\sqrt [3]{-1+\sqrt {17}}+\sqrt [3]{3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}-\frac {\sqrt [3]{71+17 \sqrt {17}} \text {Subst}\left (\int \frac {-2 \sqrt [3]{-1+\sqrt {17}}-\sqrt [3]{3+\sqrt {17}} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+\sqrt {17}}-\sqrt [3]{-3+\sqrt {17}} x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}+\frac {\sqrt [3]{-289+71 \sqrt {17}} \text {Subst}\left (\int \frac {2 \sqrt [3]{1+\sqrt {17}}+\sqrt [3]{-3+\sqrt {17}} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{459-109 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{34^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \text {Subst}\left (\int \frac {\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )}+2 \left (-3+\sqrt {17}\right )^{2/3} x}{\left (1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-3+\sqrt {17}\right ) \left (1+\sqrt {17}\right )} x+\left (-3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {Subst}\left (\int \frac {\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )}+2 \left (3+\sqrt {17}\right )^{2/3} x}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}+\frac {\sqrt [3]{109+27 \sqrt {17}} \text {Subst}\left (\int \frac {1}{\left (-1+\sqrt {17}\right )^{2/3}+\sqrt [3]{\left (-1+\sqrt {17}\right ) \left (3+\sqrt {17}\right )} x+\left (3+\sqrt {17}\right )^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )}{2^{2/3} \sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{10-2 \sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{\sqrt {17}}\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3} 17^{2/3}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2 \left (5+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {51}}+\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\left (1+\sqrt {17}\right )^{2/3}+\frac {\left (-3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7-\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (85-19 \sqrt {17}\right )} \log \left (\sqrt [3]{1+\sqrt {17}}-\frac {\sqrt [3]{-3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3\ 17^{2/3}}-\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\sqrt [3]{-1+\sqrt {17}}-\frac {\sqrt [3]{3+\sqrt {17}} x}{\sqrt [3]{1+x^3}}\right )}{3 \sqrt {17}}+\frac {\sqrt [3]{\frac {1}{2} \left (19+5 \sqrt {17}\right )} \log \left (\left (-1+\sqrt {17}\right )^{2/3}+\frac {\left (3+\sqrt {17}\right )^{2/3} x^2}{\left (1+x^3\right )^{2/3}}+\frac {\sqrt [3]{2 \left (7+\sqrt {17}\right )} x}{\sqrt [3]{1+x^3}}\right )}{6 \sqrt {17}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 103, normalized size = 1.00 \begin {gather*} -\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{3} \text {RootSum}\left [1-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$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 = 274.94, size = 7288, normalized size = 70.76
method | result | size |
risch | \(\text {Expression too large to display}\) | \(7288\) |
trager | \(\text {Expression too large to display}\) | \(10402\) |
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 )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (2\,x^6+x^3-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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