Optimal. Leaf size=32 \[ -\frac {1}{2 (x+1)}+\frac {1}{4} \log (1-x)-\log (x)+\frac {3}{4} \log (x+1) \]
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Rubi [A] time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {72} \[ -\frac {1}{2 (x+1)}+\frac {1}{4} \log (1-x)-\log (x)+\frac {3}{4} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 72
Rubi steps
\begin {align*} \int \frac {1}{(-1+x) x (1+x)^2} \, dx &=\int \left (\frac {1}{4 (-1+x)}-\frac {1}{x}+\frac {1}{2 (1+x)^2}+\frac {3}{4 (1+x)}\right ) \, dx\\ &=-\frac {1}{2 (1+x)}+\frac {1}{4} \log (1-x)-\log (x)+\frac {3}{4} \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.88 \[ \frac {1}{4} \left (-\frac {2}{x+1}+\log (1-x)-4 \log (x)+3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 33, normalized size = 1.03 \[ \frac {3 \, {\left (x + 1\right )} \log \left (x + 1\right ) + {\left (x + 1\right )} \log \left (x - 1\right ) - 4 \, {\left (x + 1\right )} \log \relax (x) - 2}{4 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 34, normalized size = 1.06 \[ -\frac {1}{2 \, {\left (x + 1\right )}} - \log \left ({\left | -\frac {1}{x + 1} + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | -\frac {2}{x + 1} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.78 \[ -\ln \relax (x )+\frac {\ln \left (x -1\right )}{4}+\frac {3 \ln \left (x +1\right )}{4}-\frac {1}{2 \left (x +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 24, normalized size = 0.75 \[ -\frac {1}{2 \, {\left (x + 1\right )}} + \frac {3}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 26, normalized size = 0.81 \[ \frac {\ln \left (x-1\right )}{4}+\frac {3\,\ln \left (x+1\right )}{4}-\ln \relax (x)-\frac {1}{2\,\left (x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 24, normalized size = 0.75 \[ - \log {\relax (x )} + \frac {\log {\left (x - 1 \right )}}{4} + \frac {3 \log {\left (x + 1 \right )}}{4} - \frac {1}{2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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