Optimal. Leaf size=112 \[ \frac {\left (x^3-1\right )^{2/3} \left (9 x^3-4\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [4 \text {$\#$1}^6-10 \text {$\#$1}^3+7\& ,\frac {-6 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+6 \text {$\#$1}^3 \log (x)+7 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-7 \log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ] \]
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Rubi [C] time = 0.52, antiderivative size = 239, normalized size of antiderivative = 2.13, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 264, 277, 239, 430, 429} \begin {gather*} \frac {\left (\sqrt {3}+3 i\right ) 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}{1-i \sqrt {3}}\right )}{12 \left (\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (-\sqrt {3}+3 i\right ) 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}{1+i \sqrt {3}}\right )}{12 \left (-\sqrt {3}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{4} \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 )}{2 \sqrt {3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 277
Rule 429
Rule 430
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{2 \left (4+2 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 (2+x^3\right )}{4+2 x^3+x^6} \, dx+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\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}{6} \left (3-i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\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 (3-i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx}{6 \left (1-x^3\right )^{2/3}}+\frac {\left (\left (3+i \sqrt {3}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx}{6 \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (3 i+\sqrt {3}\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 {x^3}{1-i \sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (3 i-\sqrt {3}\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 {x^3}{1+i \sqrt {3}}\right )}{12 \left (i-\sqrt {3}\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 [C] time = 0.31, size = 641, normalized size = 5.72 \begin {gather*} \left (x^3-1\right )^{2/3} \left (\frac {9}{20 x^2}-\frac {1}{5 x^5}\right )+\frac {1}{72} \left (\frac {2 \sqrt [3]{\frac {\sqrt {3}-i}{\sqrt {3}-2 i}} \left (5 \sqrt {3}-3 i\right ) \log \left (\sqrt [3]{-\sqrt {3}+i}-\frac {\sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}\right )}{\sqrt {3}-i}+\frac {2 \left (5 \sqrt {3}+3 i\right ) \log \left (\sqrt [3]{\sqrt {3}+i}-\frac {\sqrt [3]{\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}\right )}{\left (\sqrt {3}+i\right )^{2/3} \sqrt [3]{\sqrt {3}+2 i}}-\frac {\left (5 \sqrt {3}+3 i\right ) \left (2 \sqrt {3} \sqrt [3]{\frac {1+3 i \sqrt {3}}{\sqrt {3}+i}} \tan ^{-1}\left (\frac {1+\frac {2 \left (\sqrt {3}+2 i\right )^{2/3} x}{\sqrt [3]{1+3 i \sqrt {3}} \sqrt [3]{x^3-1}}}{\sqrt {3}}\right )+\sqrt [3]{\sqrt {3}+2 i} \log \left (\frac {\sqrt [3]{1+3 i \sqrt {3}} x}{\sqrt [3]{x^3-1}}+\frac {\left (\sqrt {3}+2 i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+\left (\sqrt {3}+i\right )^{2/3}\right )\right )}{\left (1+3 i \sqrt {3}\right )^{2/3}}-\frac {\left (5 \sqrt {3}-3 i\right ) \left (2 \sqrt {-3 \left (-\sqrt {3}+i\right )^{2/3}} \tanh ^{-1}\left (\frac {\sqrt [3]{-\sqrt {3}+i} \sqrt [3]{x^3-1}+2 \sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt {-3 \left (-\sqrt {3}+i\right )^{2/3}} \sqrt [3]{x^3-1}}\right )+\sqrt [3]{-\sqrt {3}+i} \log \left (-\frac {\sqrt [3]{-\sqrt {3}+i} \sqrt [3]{-\sqrt {3}+2 i} x}{\sqrt [3]{x^3-1}}-\frac {\left (-\sqrt {3}+2 i\right )^{2/3} x^2}{\left (x^3-1\right )^{2/3}}-\left (-\sqrt {3}+i\right )^{2/3}\right )\right )}{\sqrt [3]{-\sqrt {3}+2 i} \left (\sqrt {3}-i\right )}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3} \left (-4+9 x^3\right )}{20 x^5}-\frac {1}{12} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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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} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}\, dx\]
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} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\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+4\right )}{x^6\,\left (x^6+2\,x^3+4\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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