Optimal. Leaf size=32 \[ -\frac {3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac {1}{\sqrt [3]{x}}+1\right )-\log (x) \]
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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 = {190, 44} \[ -\frac {3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\frac {1}{\sqrt [3]{x}}+1\right )-\log (x) \]
Antiderivative was successfully verified.
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Rule 44
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{1+\frac {1}{\sqrt [3]{x}}} \, dx &=-\left (3 \operatorname {Subst}\left (\int \frac {1}{x^4 (1+x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=-\left (3 \operatorname {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 \[ -\frac {3 x^{2/3}}{2}+x+3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 20, normalized size = 0.62 \[ x - \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 20, normalized size = 0.62 \[ x - \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, x^{\frac {1}{3}} - 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.66 \[ x -3 \ln \left (x^{\frac {1}{3}}+1\right )-\frac {3 x^{\frac {2}{3}}}{2}+3 x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 28, normalized size = 0.88 \[ -\frac {1}{2} \, x {\left (\frac {3}{x^{\frac {1}{3}}} - \frac {6}{x^{\frac {2}{3}}} - 2\right )} - \log \relax (x) - 3 \, \log \left (\frac {1}{x^{\frac {1}{3}}} + 1\right ) \]
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 \[ x-3\,\ln \left (x^{1/3}+1\right )+3\,x^{1/3}-\frac {3\,x^{2/3}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 26, normalized size = 0.81 \[ - \frac {3 x^{\frac {2}{3}}}{2} + 3 \sqrt [3]{x} + x - 3 \log {\left (\sqrt [3]{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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