Optimal. Leaf size=26 \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]
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Rubi [A] time = 0.0159524, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {376, 77} \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 376
Rule 77
Rubi steps
\begin{align*} \int \frac{-1+\sqrt [3]{x}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname{Subst}\left (\int \frac{(-1+x) x^2}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (2-2 x+x^2-\frac{2}{1+x}\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*}
Mathematica [A] time = 0.0103988, size = 26, normalized size = 1. \[ -3 x^{2/3}+x+6 \sqrt [3]{x}-6 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.8 \begin{align*} 6\,\sqrt [3]{x}-3\,{x}^{2/3}+x-6\,\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 = 0.940811, size = 27, normalized size = 1.04 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (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 = 1.3296, size = 65, normalized size = 2.5 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \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 = 0.166887, size = 24, normalized size = 0.92 \begin{align*} - 3 x^{\frac{2}{3}} + 6 \sqrt [3]{x} + x - 6 \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 = 1.10426, size = 27, normalized size = 1.04 \begin{align*} x - 3 \, x^{\frac{2}{3}} + 6 \, x^{\frac{1}{3}} - 6 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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