Optimal. Leaf size=110 \[ \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\& ,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.33, antiderivative size = 281, normalized size of antiderivative = 2.55, number of steps
used = 13, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {6857, 270,
283, 245, 399, 384} \begin {gather*} \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) (2-i)^{2/3} \text {ArcTan}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{2-i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) (2+i)^{2/3} \text {ArcTan}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{2+i} x}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\left (\frac {1}{48}-\frac {i}{48}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-x^3+2 i\right )+\left (\frac {1}{48}+\frac {i}{48}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (x^3+2 i\right )-\left (\frac {1}{16}+\frac {i}{16}\right ) \left (1-\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1-\frac {i}{2}} x\right )-\left (\frac {1}{16}-\frac {i}{16}\right ) \left (1+\frac {i}{2}\right )^{2/3} \log \left (-\sqrt [3]{x^3-1}+\sqrt [3]{1+\frac {i}{2}} x\right )+\frac {\left (x^3-1\right )^{5/3}}{10 x^5}-\frac {\left (x^3-1\right )^{2/3}}{8 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 245
Rule 270
Rule 283
Rule 384
Rule 399
Rule 6857
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (4+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}}{4 x^3}+\frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4 \left (4+x^6\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+\frac {1}{4} \int \frac {\left (-2-x^3\right ) \left (-1+x^3\right )^{2/3}}{4+x^6} \, dx+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{4} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{4} \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i-x^3}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-1+x^3\right )^{2/3}}{2 i+x^3}\right ) \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 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 )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i+x^3} \, dx+\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{2 i-x^3} \, dx\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 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 )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )+-\frac {\left (\left (\frac {1}{8}+\frac {i}{8}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2 i+x^3} \, dx}{\left (1-x^3\right )^{2/3}}+\frac {\left (\left (\frac {1}{8}-\frac {i}{8}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{2 i-x^3} \, dx}{\left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{2/3}}{8 x^2}+\frac {\left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {\left (\frac {1}{16}-\frac {i}{16}\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 {i x^3}{2}\right )}{\left (1-x^3\right )^{2/3}}-\frac {\left (\frac {1}{16}+\frac {i}{16}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};1,-\frac {2}{3};\frac {4}{3};-\frac {i x^3}{2},x^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 )}{4 \sqrt {3}}-\frac {1}{8} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 110, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \left (-1+x^3\right )^{2/3}}{40 x^5}+\frac {1}{96} \text {RootSum}\left [5-8 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$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 = 28.56, size = 8127, normalized size = 73.88 \[\text {output too large to display}\]
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]
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^{3} + 2\right )}{x^{6} \left (x^{6} + 4\right )}\, dx \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 (x^6+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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