Optimal. Leaf size=30 \[ -6 \sqrt [3]{x}-3 x^{2/3}-x-6 \log \left (1-\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {381, 383, 78}
\begin {gather*} -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 381
Rule 383
Rubi steps
\begin {align*} \int \frac {1+\frac {1}{\sqrt [3]{x}}}{-1+\frac {1}{\sqrt [3]{x}}} \, dx &=\int \frac {1+\sqrt [3]{x}}{1-\sqrt [3]{x}} \, dx\\ &=3 \text {Subst}\left (\int \frac {x^2 (1+x)}{1-x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (-2-\frac {2}{-1+x}-2 x-x^2\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-6 \sqrt [3]{x}-3 x^{2/3}-x-6 \log \left (1-\sqrt [3]{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.93 \begin {gather*} -6 \sqrt [3]{x}-3 x^{2/3}-x-6 \log \left (-1+\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 23, normalized size = 0.77
method | result | size |
derivativedivides | \(-x -3 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}-6 \ln \left (x^{\frac {1}{3}}-1\right )\) | \(23\) |
default | \(-x -3 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}-6 \ln \left (x^{\frac {1}{3}}-1\right )\) | \(23\) |
trager | \(2-x -6 x^{\frac {1}{3}}-3 x^{\frac {2}{3}}-2 \ln \left (-3 x^{\frac {2}{3}}+3 x^{\frac {1}{3}}+x -1\right )\) | \(32\) |
meijerg | \(-\frac {x^{\frac {1}{3}} \left (4 x^{\frac {2}{3}}+6 x^{\frac {1}{3}}+12\right )}{4}-6 \ln \left (1-x^{\frac {1}{3}}\right )-\frac {x^{\frac {1}{3}} \left (6+3 x^{\frac {1}{3}}\right )}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 0.73 \begin {gather*} -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.15, size = 22, normalized size = 0.73 \begin {gather*} -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 0.87 \begin {gather*} - 3 x^{\frac {2}{3}} - 6 \sqrt [3]{x} - x - 6 \log {\left (\sqrt [3]{x} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 23, normalized size = 0.77 \begin {gather*} -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.73 \begin {gather*} -x-6\,\ln \left (x^{1/3}-1\right )-6\,x^{1/3}-3\,x^{2/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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