Optimal. Leaf size=28 \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]
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Rubi [A] time = 0.0111347, antiderivative size = 28, normalized size of antiderivative = 1., 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, 88} \[ \frac{1}{2 (x+1)}+\frac{1}{4} \log (1-x)+\frac{3}{4} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 27
Rule 88
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*}
Mathematica [A] time = 0.0141945, size = 22, normalized size = 0.79 \[ \frac{1}{4} \left (\frac{2}{x+1}+\log (x-1)+3 \log (x+1)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 21, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x-1 \right ) }{4}}+{\frac{1}{2+2\,x}}+{\frac{3\,\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 = 1.0184, size = 27, normalized size = 0.96 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72075, size = 82, normalized size = 2.93 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104101, size = 20, normalized size = 0.71 \begin{align*} \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 = 1.15438, size = 30, normalized size = 1.07 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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