Optimal. Leaf size=20 \[ 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 272, 45}
\begin {gather*} 3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{-\sqrt [3]{x}+x^{2/3}} \, dx &=\int \frac {1}{\left (-1+\sqrt [3]{x}\right ) \sqrt [3]{x}} \, dx\\ &=3 \text {Subst}\left (\int \frac {x}{-1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=3 \sqrt [3]{x}+3 \log \left (1-\sqrt [3]{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.90 \begin {gather*} 3 \sqrt [3]{x}+3 \log \left (-1+\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 15, normalized size = 0.75
method | result | size |
derivativedivides | \(3 x^{\frac {1}{3}}+3 \ln \left (-1+x^{\frac {1}{3}}\right )\) | \(15\) |
default | \(3 x^{\frac {1}{3}}+3 \ln \left (-1+x^{\frac {1}{3}}\right )\) | \(15\) |
meijerg | \(3 x^{\frac {1}{3}}+3 \ln \left (1-x^{\frac {1}{3}}\right )\) | \(17\) |
trager | \(3 x^{\frac {1}{3}}+\ln \left (-3 x^{\frac {2}{3}}+3 x^{\frac {1}{3}}+x -1\right )\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 14, normalized size = 0.70 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \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 = 0.34, size = 14, normalized size = 0.70 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \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 = 15, normalized size = 0.75 \begin {gather*} 3 \sqrt [3]{x} + 3 \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 = 4.53, size = 15, normalized size = 0.75 \begin {gather*} 3 \, x^{\frac {1}{3}} + 3 \, \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 = 0.08, size = 14, normalized size = 0.70 \begin {gather*} 3\,\ln \left (x^{1/3}-1\right )+3\,x^{1/3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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