Optimal. Leaf size=16 \[ \frac {1}{2 (x+1)}+\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {77, 207} \[ \frac {1}{2 (x+1)}+\frac {1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 77
Rule 207
Rubi steps
\begin {align*} \int \frac {x}{(1-x) (1+x)^2} \, dx &=\int \left (-\frac {1}{2 (1+x)^2}-\frac {1}{2 \left (-1+x^2\right )}\right ) \, dx\\ &=\frac {1}{2 (1+x)}-\frac {1}{2} \int \frac {1}{-1+x^2} \, dx\\ &=\frac {1}{2 (1+x)}+\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.50 \[ \frac {1}{4} \left (\frac {2}{x+1}-\log (1-x)+\log (x+1)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 26, normalized size = 1.62 \[ \frac {{\left (x + 1\right )} \log \left (x + 1\right ) - {\left (x + 1\right )} \log \left (x - 1\right ) + 2}{4 \, {\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 21, normalized size = 1.31 \[ \frac {1}{2 \, {\left (x + 1\right )}} - \frac {1}{4} \, \log \left ({\left | -\frac {2}{x + 1} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 1.31 \[ -\frac {\ln \left (x -1\right )}{4}+\frac {\ln \left (x +1\right )}{4}+\frac {1}{2 x +2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 20, normalized size = 1.25 \[ \frac {1}{2 \, {\left (x + 1\right )}} + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 12, normalized size = 0.75 \[ \frac {\mathrm {atanh}\relax (x)}{2}+\frac {1}{2\,\left (x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 19, normalized size = 1.19 \[ - \frac {\log {\left (x - 1 \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{4} + \frac {1}{2 x + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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