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 = 0.0132232, antiderivative size = 32, normalized size of antiderivative = 1., 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 190
Rule 44
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*}
Mathematica [A] time = 0.010988, 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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Maple [A] time = 0.002, size = 21, normalized size = 0.7 \begin{align*} x-{\frac{3}{2}{x}^{{\frac{2}{3}}}}+3\,\sqrt [3]{x}-3\,\ln \left ( \sqrt [3]{x}+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.930783, size = 38, normalized size = 1.19 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90929, size = 68, normalized size = 2.12 \begin{align*} x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.113406, size = 26, normalized size = 0.81 \begin{align*} - \frac{3 x^{\frac{2}{3}}}{2} + 3 \sqrt [3]{x} + x - 3 \log{\left (\sqrt [3]{x} + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04935, size = 27, normalized size = 0.84 \begin{align*} x - \frac{3}{2} \, x^{\frac{2}{3}} + 3 \, x^{\frac{1}{3}} - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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