Optimal. Leaf size=109 \[ \frac {1}{24} \text {RootSum}\left [2 \text {$\#$1}^6-4 \text {$\#$1}^3+1\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+2 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-2 \log (x)}{\text {$\#$1}^4-\text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (1-4 x^3\right )}{10 x^5} \]
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Rubi [A] time = 0.51, antiderivative size = 157, normalized size of antiderivative = 1.44, 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 = {28, 6725, 264, 277, 239, 429} \begin {gather*} -\frac {1}{8} \left (3+2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{\sqrt {2}}\right )-\frac {1}{8} \left (3-2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};\frac {x^3}{\sqrt {2}},-x^3\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 28
Rule 239
Rule 264
Rule 277
Rule 429
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx &=\int \frac {\left (-1+x^3\right )^2 \left (1+x^3\right )^{2/3}}{x^6 \left (-2+x^6\right )} \, dx\\ &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (-2+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\right )+\frac {1}{2} \int \frac {\left (3-2 x^3\right ) \left (1+x^3\right )^{2/3}}{-2+x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{2} \int \left (-\frac {\left (3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}-x^3\right )}+\frac {\left (-3-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (\sqrt {2}+x^3\right )}\right ) \, dx+\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\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}{8} \left (4-3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}-x^3} \, dx-\frac {1}{8} \left (4+3 \sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{\sqrt {2}+x^3} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{10 x^5}-\frac {1}{8} \left (3+2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{\sqrt {2}}\right )-\frac {1}{8} \left (3-2 \sqrt {2}\right ) x F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};\frac {x^3}{\sqrt {2}},-x^3\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 [F] time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+x^6\right )}{x^6 \left (-2+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.00, size = 109, normalized size = 1.00 \begin {gather*} \frac {\left (1-4 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+\frac {1}{24} \text {RootSum}\left [1-4 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\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} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 239.11, size = 6590, normalized size = 60.46
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6590\) |
trager | \(\text {Expression too large to display}\) | \(9174\) |
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} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 2\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+1\right )}{x^6\,\left (x^6-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}{x^{6} \left (x^{6} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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