Optimal. Leaf size=32 \[ 3 \sqrt [3]{x}-\frac {3 x^{2/3}}{2}+x-3 \log \left (1+\frac {1}{\sqrt [3]{x}}\right )-\log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {196, 46}
\begin {gather*} -\frac {3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac {1}{\sqrt [3]{x}}+1\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 196
Rubi steps
\begin {align*} \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx &=-\left (3 \text {Subst}\left (\int \frac {1}{x^4 (1+x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \text {Subst}\left (\int \left (\frac {1}{x^4}-\frac {1}{x^3}+\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=3 \sqrt [3]{x}-\frac {3 x^{2/3}}{2}+x-3 \log \left (1+\frac {1}{\sqrt [3]{x}}\right )-\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 0.88 \begin {gather*} 3 \sqrt [3]{x}-\frac {3 x^{2/3}}{2}+x-3 \log \left (1+\sqrt [3]{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.75, size = 20, normalized size = 0.62 \begin {gather*} 3 x^{\frac {1}{3}}-\frac {3 x^{\frac {2}{3}}}{2}+x-3 \text {Log}\left [1+x^{\frac {1}{3}}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 21, normalized size = 0.66
method | result | size |
derivativedivides | \(x -\frac {3 x^{\frac {2}{3}}}{2}+3 x^{\frac {1}{3}}-3 \ln \left (x^{\frac {1}{3}}+1\right )\) | \(21\) |
default | \(x -\frac {3 x^{\frac {2}{3}}}{2}+3 x^{\frac {1}{3}}-3 \ln \left (x^{\frac {1}{3}}+1\right )\) | \(21\) |
meijerg | \(\frac {x^{\frac {1}{3}} \left (4 x^{\frac {2}{3}}-6 x^{\frac {1}{3}}+12\right )}{4}-3 \ln \left (x^{\frac {1}{3}}+1\right )\) | \(27\) |
trager | \(-1+x +3 x^{\frac {1}{3}}-\frac {3 x^{\frac {2}{3}}}{2}-\ln \left (-3 x^{\frac {2}{3}}-3 x^{\frac {1}{3}}-x -1\right )\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 28, normalized size = 0.88 \begin {gather*} -\frac {1}{2} \, x {\left (\frac {3}{x^{\frac {1}{3}}} - \frac {6}{x^{\frac {2}{3}}} - 2\right )} - \log \left (x\right ) - 3 \, \log \left (\frac {1}{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 = 20, normalized size = 0.62 \begin {gather*} x - \frac {3}{2} \, x^{\frac {2}{3}} + 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 = 0.06, size = 26, normalized size = 0.81 \begin {gather*} - \frac {3 x^{\frac {2}{3}}}{2} + 3 \sqrt [3]{x} + 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 = 0.00, size = 33, normalized size = 1.03 \begin {gather*} 3 \left (\frac {x}{3}-\frac {\left (x^{\frac {1}{3}}\right )^{2}}{2}+x^{\frac {1}{3}}-\ln \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 = 0.03, size = 20, normalized size = 0.62 \begin {gather*} x-3\,\ln \left (x^{1/3}+1\right )+3\,x^{1/3}-\frac {3\,x^{2/3}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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