Optimal. Leaf size=270 \[ -\frac {\left (1-x^3\right )^{2/3}}{x}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\log \left ((1-x) (1+x)^2\right )}{12 \sqrt [3]{2}}-\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {491, 598,
371, 502, 2174, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )-\frac {\left (1-x^3\right )^{2/3}}{x}-\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}+\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}-\frac {\log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 371
Rule 491
Rule 502
Rule 598
Rule 631
Rule 642
Rule 648
Rule 2174
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=-\frac {F_1\left (-\frac {1}{3};\frac {1}{3},1;\frac {2}{3};x^3,-x^3\right )}{x}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 20.04, size = 67, normalized size = 0.25 \begin {gather*} -\frac {\left (1-x^3\right )^{2/3}}{x}-x^2 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^3,-x^3\right )-\frac {1}{5} x^5 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};x^3,-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+1\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\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 = -0.00 \begin {gather*} \int \frac {1}{x^2\,{\left (1-x^3\right )}^{1/3}\,\left (x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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