Optimal. Leaf size=232 \[ \sqrt [3]{1-x^3}+\frac {1}{2} x F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-\frac {x^3}{8}\right )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {1-\frac {3^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )-\sqrt [6]{3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1-x^3}}{3 \sqrt [6]{3}}\right )-\frac {\log \left (8+x^3\right )}{\sqrt [3]{3}}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{1-x^3}\right )-\log \left (-x-\sqrt [3]{1-x^3}\right )+\frac {1}{2} 3^{2/3} \log \left (-\frac {1}{2} 3^{2/3} x-\sqrt [3]{1-x^3}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {2181, 440,
495, 337, 503, 455, 52, 59, 631, 210, 31} \begin {gather*} \frac {1}{2} x F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-\frac {x^3}{8}\right )+\sqrt [3]{1-x^3}-\frac {\log \left (x^3+8\right )}{\sqrt [3]{3}}+\frac {1}{2} 3^{2/3} \log \left (3^{2/3}-\sqrt [3]{1-x^3}\right )-\log \left (-\sqrt [3]{1-x^3}-x\right )+\frac {1}{2} 3^{2/3} \log \left (-\sqrt [3]{1-x^3}-\frac {1}{2} 3^{2/3} x\right )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\sqrt [6]{3} \tan ^{-1}\left (\frac {1-\frac {3^{2/3} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )-\sqrt [6]{3} \tan ^{-1}\left (\frac {2 \sqrt [3]{1-x^3}}{3 \sqrt [6]{3}}+\frac {1}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 59
Rule 210
Rule 337
Rule 440
Rule 455
Rule 495
Rule 503
Rule 631
Rule 2181
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx &=\int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx\\ \end {align*}
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Mathematica [F]
time = 33.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1-x^3}}{2+x} \, dx \end {gather*}
Verification is not applicable to the result.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{3}+1\right )^{\frac {1}{3}}}{2+x}\, dx\]
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(-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 {\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )}}{x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 {{\left (1-x^3\right )}^{1/3}}{x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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