Optimal. Leaf size=86 \[ \frac {\left (1-x^3\right ) \left (x^3-1\right )^{2/3}}{10 x^5}-\frac {1}{8} \text {RootSum}\left [8 \text {$\#$1}^6-12 \text {$\#$1}^3+3\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{4 \text {$\#$1}^3-3}\& \right ] \]
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Rubi [A] time = 0.48, antiderivative size = 159, normalized size of antiderivative = 1.85, number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 264, 1428, 430, 429} \begin {gather*} \frac {\sqrt {3} x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2 \left (1-\sqrt {3}\right )}\right )}{16 \left (1-\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\sqrt {3} x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2 \left (1+\sqrt {3}\right )}\right )}{16 \left (1+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{10 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 264
Rule 429
Rule 430
Rule 1428
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (-1+x^3\right )^{2/3}}{2 \left (-8+4 x^3+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\right )+\frac {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-8+4 x^3+x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \sqrt {3} \int \frac {\left (-1+x^3\right )^{2/3}}{4-4 \sqrt {3}+2 x^3} \, dx-\frac {1}{4} \sqrt {3} \int \frac {\left (-1+x^3\right )^{2/3}}{4+4 \sqrt {3}+2 x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\left (\sqrt {3} \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{4-4 \sqrt {3}+2 x^3} \, dx}{4 \left (1-x^3\right )^{2/3}}-\frac {\left (\sqrt {3} \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{4+4 \sqrt {3}+2 x^3} \, dx}{4 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {\sqrt {3} x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2 \left (1-\sqrt {3}\right )}\right )}{16 \left (1-\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}-\frac {\sqrt {3} x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2 \left (1+\sqrt {3}\right )}\right )}{16 \left (1+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4-2 x^3+x^6\right )}{x^6 \left (-8+4 x^3+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.20, size = 86, normalized size = 1.00 \begin {gather*} \frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{10 x^5}-\frac {1}{8} \text {RootSum}\left [3-12 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+4 \text {$\#$1}^3}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} - 2 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 8\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 374.27, size = 6935, normalized size = 80.64
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6935\) |
trager | \(\text {Expression too large to display}\) | \(12165\) |
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*} \int \frac {{\left (x^{6} - 2 \, x^{3} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 4 \, x^{3} - 8\right )} x^{6}}\,{d x} \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^6-2\,x^3+4\right )}{x^6\,\left (x^6+4\,x^3-8\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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