Optimal. Leaf size=30 \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]
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Rubi [A] time = 0.02, 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 = {374, 376, 77} \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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Rule 77
Rule 374
Rule 376
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 \operatorname {Subst}\left (\int \frac {x^2 (1+x)}{1-x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {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 = 0.01, size = 30, normalized size = 1.00 \[ -3 x^{2/3}-x-6 \sqrt [3]{x}-6 \log \left (1-\sqrt [3]{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 22, normalized size = 0.73 \[ -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 23, normalized size = 0.77 \[ -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left ({\left | x^{\frac {1}{3}} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.77 \[ -x -6 \ln \left (x^{\frac {1}{3}}-1\right )-3 x^{\frac {2}{3}}-6 x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 22, normalized size = 0.73 \[ -x - 3 \, x^{\frac {2}{3}} - 6 \, x^{\frac {1}{3}} - 6 \, \log \left (x^{\frac {1}{3}} - 1\right ) \]
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 \[ -x-6\,\ln \left (x^{1/3}-1\right )-6\,x^{1/3}-3\,x^{2/3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 26, normalized size = 0.87 \[ - 3 x^{\frac {2}{3}} - 6 \sqrt [3]{x} - x - 6 \log {\left (\sqrt [3]{x} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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