Optimal. Leaf size=79 \[ \frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2}{3} \text {RootSum}\left [5-5 \text {$\#$1}^3+2 \text {$\#$1}^6\& ,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-5+4 \text {$\#$1}^3}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.77, antiderivative size = 455, normalized size of antiderivative = 5.76, number of steps
used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6860, 270,
1442, 399, 245, 384} \begin {gather*} \frac {2 i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {-\sqrt {15}+i}{-\sqrt {15}+5 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {5} \left (\frac {-\sqrt {15}+i}{-\sqrt {15}+5 i}\right )^{2/3}}-\frac {2 i \text {ArcTan}\left (\frac {1+\frac {2 x}{\sqrt [3]{\frac {\sqrt {15}+i}{\sqrt {15}+5 i}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {5} \left (\frac {\sqrt {15}+i}{\sqrt {15}+5 i}\right )^{2/3}}-\frac {i \log \left (4 x^3-i \sqrt {15}+1\right )}{3 \sqrt {15} \left (\frac {\sqrt {15}+i}{\sqrt {15}+5 i}\right )^{2/3}}+\frac {i \log \left (4 x^3+i \sqrt {15}+1\right )}{3 \sqrt {15} \left (\frac {-\sqrt {15}+i}{-\sqrt {15}+5 i}\right )^{2/3}}-\frac {i \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {-\sqrt {15}+i}{-\sqrt {15}+5 i}}}\right )}{\sqrt {15} \left (\frac {-\sqrt {15}+i}{-\sqrt {15}+5 i}\right )^{2/3}}+\frac {i \log \left (-\sqrt [3]{x^3-1}+\frac {x}{\sqrt [3]{\frac {\sqrt {15}+i}{\sqrt {15}+5 i}}}\right )}{\sqrt {15} \left (\frac {\sqrt {15}+i}{\sqrt {15}+5 i}\right )^{2/3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 270
Rule 384
Rule 399
Rule 1442
Rule 6860
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {2 \left (-1+x^3\right )^{2/3}}{2+x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\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^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {(8 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1-i \sqrt {15}+4 x^3} \, dx}{\sqrt {15}}-\frac {(8 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {15}+4 x^3} \, dx}{\sqrt {15}}\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (8 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-i \sqrt {15}+4 x^3} \, dx}{\sqrt {15} \left (1-x^3\right )^{2/3}}-\frac {\left (8 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+i \sqrt {15}+4 x^3} \, dx}{\sqrt {15} \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {8 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-i \sqrt {15}}\right )}{\sqrt {15} \left (i+\sqrt {15}\right ) \left (1-x^3\right )^{2/3}}+\frac {8 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+i \sqrt {15}}\right )}{\sqrt {15} \left (i-\sqrt {15}\right ) \left (1-x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 79, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2}{3} \text {RootSum}\left [5-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-5+4 \text {$\#$1}^3}\&\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 = 192.95, size = 7735, normalized size = 97.91
method | result | size |
risch | \(\text {Expression too large to display}\) | \(7735\) |
trager | \(\text {Expression too large to display}\) | \(10913\) |
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^6\,\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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