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 = 0.0123629, antiderivative size = 32, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0107612, 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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Maple [A] time = 0.008, size = 25, normalized size = 0.8 \begin{align*} -\ln \left ( x \right ) -{\frac{1}{2+2\,x}}+{\frac{3\,\ln \left ( 1+x \right ) }{4}}+{\frac{\ln \left ( -1+x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948003, size = 32, normalized size = 1. \begin{align*} -\frac{1}{2 \,{\left (x + 1\right )}} + \frac{3}{4} \, \log \left (x + 1\right ) + \frac{1}{4} \, \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77226, size = 108, normalized size = 3.38 \begin{align*} \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 \left (x\right ) - 2}{4 \,{\left (x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.134233, size = 24, normalized size = 0.75 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x - 1 \right )}}{4} + \frac{3 \log{\left (x + 1 \right )}}{4} - \frac{1}{2 x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08221, size = 46, normalized size = 1.44 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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