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 = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 628
Rule 1834
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*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ -\frac {1}{2 x^2}+\frac {1}{3} \log \left (x^2-x+1\right )+\frac {1}{x}-\frac {2}{3} \log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 33, normalized size = 1.03 \[ \frac {2 \, x^{2} \log \left (x^{2} - x + 1\right ) - 4 \, x^{2} \log \left (x + 1\right ) + 6 \, x - 3}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.91 \[ \frac {2 \, x - 1}{2 \, x^{2}} + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left ({\left | x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 27, normalized size = 0.84 \[ -\frac {2 \ln \left (x +1\right )}{3}+\frac {\ln \left (x^{2}-x +1\right )}{3}+\frac {1}{x}-\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 28, normalized size = 0.88 \[ \frac {2 \, x - 1}{2 \, x^{2}} + \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {2}{3} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 25, normalized size = 0.78 \[ \frac {\ln \left (x^2-x+1\right )}{3}-\frac {2\,\ln \left (x+1\right )}{3}+\frac {x-\frac {1}{2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 27, normalized size = 0.84 \[ - \frac {2 \log {\left (x + 1 \right )}}{3} + \frac {\log {\left (x^{2} - x + 1 \right )}}{3} - \frac {1 - 2 x}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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