Optimal. Leaf size=28 \[ \frac {1}{2 (1+x)}+\frac {1}{4} \log (1-x)+\frac {3}{4} \log (1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 90}
\begin {gather*} \frac {1}{2 (x+1)}+\frac {1}{4} \log (1-x)+\frac {3}{4} \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 90
Rubi steps
\begin {align*} \int \frac {x^2}{(-1+x) \left (1+2 x+x^2\right )} \, dx &=\int \frac {x^2}{(-1+x) (1+x)^2} \, dx\\ &=\int \left (\frac {1}{4 (-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)+\frac {3}{4} \log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{4} \left (\frac {2}{1+x}+\log (-1+x)+3 \log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 21, normalized size = 0.75
method | result | size |
default | \(\frac {\ln \left (-1+x \right )}{4}+\frac {1}{2 x +2}+\frac {3 \ln \left (1+x \right )}{4}\) | \(21\) |
norman | \(\frac {\ln \left (-1+x \right )}{4}+\frac {1}{2 x +2}+\frac {3 \ln \left (1+x \right )}{4}\) | \(21\) |
risch | \(\frac {\ln \left (-1+x \right )}{4}+\frac {1}{2 x +2}+\frac {3 \ln \left (1+x \right )}{4}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 20, normalized size = 0.71 \begin {gather*} \frac {1}{2 \, {\left (x + 1\right )}} + \frac {3}{4} \, \log \left (x + 1\right ) + \frac {1}{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 = 0.39, size = 26, normalized size = 0.93 \begin {gather*} \frac {3 \, {\left (x + 1\right )} \log \left (x + 1\right ) + {\left (x + 1\right )} \log \left (x - 1\right ) + 2}{4 \, {\left (x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x - 1 \right )}}{4} + \frac {3 \log {\left (x + 1 \right )}}{4} + \frac {1}{2 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.85, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{2 \, {\left (x + 1\right )}} + \frac {3}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 20, normalized size = 0.71 \begin {gather*} \frac {\ln \left (x-1\right )}{4}+\frac {3\,\ln \left (x+1\right )}{4}+\frac {1}{2\,\left (x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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