Optimal. Leaf size=104 \[ \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\& ,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\& \right ] \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(509\) vs. \(2(104)=208\).
time = 0.65, antiderivative size = 509, normalized size of antiderivative = 4.89, number of steps
used = 12, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6860, 283,
245, 399, 384} \begin {gather*} \frac {\left (5+\sqrt {17}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\left (5-\sqrt {17}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4 \sqrt {51}}-\frac {\text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \text {ArcTan}\left (\frac {\frac {\sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{4\ 2^{2/3} \sqrt {51}}-\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (2 x^3-\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (2 x^3+\sqrt {17}+1\right )}{24\ 2^{2/3} \sqrt {17}}+\frac {\sqrt [3]{199+47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7-\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {\sqrt [3]{199-47 \sqrt {17}} \log \left (\frac {1}{2} \sqrt [3]{7+\sqrt {17}} x-\sqrt [3]{x^3-1}\right )}{8\ 2^{2/3} \sqrt {17}}-\frac {1}{136} \left (17+5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {1}{136} \left (17-5 \sqrt {17}\right ) \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {\left (x^3-1\right )^{2/3}}{4 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 )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{2 \left (-4+x^3+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{-4+x^3+x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1+\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3}+\frac {\left (1-\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3}\right ) \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (17-5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx+\frac {1}{34} \left (17+5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (17-5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}+\frac {\left (\left (17+5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (17+5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (17-5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$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 = 60.14, size = 9277, normalized size = 89.20 \[\text {output too large to display}\]
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 (x^6+x^3-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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