Optimal. Leaf size=32 \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]
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Rubi [A] time = 0.0326286, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1834, 628} \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1834
Rule 628
Rubi steps
\begin{align*} \int \frac{1-x}{x^3 \left (1+x^3\right )} \, dx &=\int \left (\frac{1}{x^3}-\frac{1}{x^2}-\frac{2}{3 (1+x)}+\frac{-1+2 x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=-\frac{1}{2 x^2}+\frac{1}{x}-\frac{2}{3} \log (1+x)+\frac{1}{3} \int \frac{-1+2 x}{1-x+x^2} \, dx\\ &=-\frac{1}{2 x^2}+\frac{1}{x}-\frac{2}{3} \log (1+x)+\frac{1}{3} \log \left (1-x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.003987, size = 32, normalized size = 1. \[ -\frac{1}{2 x^2}+\frac{1}{3} \log \left (x^2-x+1\right )+\frac{1}{x}-\frac{2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 27, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}}}+{x}^{-1}-{\frac{2\,\ln \left ( 1+x \right ) }{3}}+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40062, size = 38, normalized size = 1.19 \begin{align*} \frac{2 \, x - 1}{2 \, x^{2}} + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42257, size = 85, normalized size = 2.66 \begin{align*} \frac{2 \, x^{2} \log \left (x^{2} - x + 1\right ) - 4 \, x^{2} \log \left (x + 1\right ) + 6 \, x - 3}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.109497, size = 27, normalized size = 0.84 \begin{align*} - \frac{2 \log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{3} + \frac{2 x - 1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04714, size = 39, normalized size = 1.22 \begin{align*} \frac{2 \, x - 1}{2 \, x^{2}} + \frac{1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac{2}{3} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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