Optimal. Leaf size=43 \[ \frac {1}{2 (1-x)}+\frac {3}{4} \log (1-x)-\frac {1}{12} \log (1+x)-\frac {1}{3} \log \left (1-x+x^2\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 43, 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 = {6857, 642}
\begin {gather*} -\frac {1}{3} \log \left (x^2-x+1\right )+\frac {1}{2 (1-x)}+\frac {3}{4} \log (1-x)-\frac {1}{12} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 6857
Rubi steps
\begin {align*} \int \frac {x^3}{(-1+x)^2 \left (1+x^3\right )} \, dx &=\int \left (\frac {1}{2 (-1+x)^2}+\frac {3}{4 (-1+x)}-\frac {1}{12 (1+x)}+\frac {1-2 x}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=\frac {1}{2 (1-x)}+\frac {3}{4} \log (1-x)-\frac {1}{12} \log (1+x)+\frac {1}{3} \int \frac {1-2 x}{1-x+x^2} \, dx\\ &=\frac {1}{2 (1-x)}+\frac {3}{4} \log (1-x)-\frac {1}{12} \log (1+x)-\frac {1}{3} \log \left (1-x+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.79 \begin {gather*} \frac {1}{12} \left (-\frac {6}{-1+x}+9 \log (-1+x)-\log (1+x)-4 \log \left ((-1+x)^2+x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 32, normalized size = 0.74
method | result | size |
default | \(-\frac {1}{2 \left (-1+x \right )}+\frac {3 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{12}-\frac {\ln \left (x^{2}-x +1\right )}{3}\) | \(32\) |
norman | \(-\frac {1}{2 \left (-1+x \right )}+\frac {3 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{12}-\frac {\ln \left (x^{2}-x +1\right )}{3}\) | \(32\) |
risch | \(-\frac {1}{2 \left (-1+x \right )}+\frac {3 \ln \left (-1+x \right )}{4}-\frac {\ln \left (1+x \right )}{12}-\frac {\ln \left (x^{2}-x +1\right )}{3}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.66, size = 31, normalized size = 0.72 \begin {gather*} -\frac {1}{2 \, {\left (x - 1\right )}} - \frac {1}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{12} \, \log \left (x + 1\right ) + \frac {3}{4} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 40, normalized size = 0.93 \begin {gather*} -\frac {4 \, {\left (x - 1\right )} \log \left (x^{2} - x + 1\right ) + {\left (x - 1\right )} \log \left (x + 1\right ) - 9 \, {\left (x - 1\right )} \log \left (x - 1\right ) + 6}{12 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 31, normalized size = 0.72 \begin {gather*} \frac {3 \log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{12} - \frac {\log {\left (x^{2} - x + 1 \right )}}{3} - \frac {1}{2 x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 36, normalized size = 0.84 \begin {gather*} -\frac {1}{2 \, {\left (x - 1\right )}} - \frac {1}{3} \, \log \left (\frac {1}{x - 1} + \frac {1}{{\left (x - 1\right )}^{2}} + 1\right ) - \frac {1}{12} \, \log \left ({\left | -\frac {2}{x - 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 33, normalized size = 0.77 \begin {gather*} \frac {3\,\ln \left (x-1\right )}{4}-\frac {\ln \left (x+1\right )}{12}-\frac {\ln \left (x^2-x+1\right )}{3}-\frac {1}{2\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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