Optimal. Leaf size=49 \[ \frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]
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Rubi [A] time = 0.0117783, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {632, 31} \[ \frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (2 x-\sqrt{5}+1\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{x}{-1+x+x^2} \, dx &=\frac{1}{10} \left (5-\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}-\frac{\sqrt{5}}{2}+x} \, dx+\frac{1}{10} \left (5+\sqrt{5}\right ) \int \frac{1}{\frac{1}{2}+\frac{\sqrt{5}}{2}+x} \, dx\\ &=\frac{1}{10} \left (5-\sqrt{5}\right ) \log \left (1-\sqrt{5}+2 x\right )+\frac{1}{10} \left (5+\sqrt{5}\right ) \log \left (1+\sqrt{5}+2 x\right )\\ \end{align*}
Mathematica [A] time = 0.0169402, size = 44, normalized size = 0.9 \[ \frac{1}{10} \left (\left (5+\sqrt{5}\right ) \log \left (2 x+\sqrt{5}+1\right )-\left (\sqrt{5}-5\right ) \log \left (-2 x+\sqrt{5}-1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 27, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}+x-1 \right ) }{2}}+{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40364, size = 50, normalized size = 1.02 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x - \sqrt{5} + 1}{2 \, x + \sqrt{5} + 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89853, size = 127, normalized size = 2.59 \begin{align*} \frac{1}{10} \, \sqrt{5} \log \left (\frac{2 \, x^{2} + \sqrt{5}{\left (2 \, x + 1\right )} + 2 \, x + 3}{x^{2} + x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} + x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.106135, size = 46, normalized size = 0.94 \begin{align*} \left (\frac{\sqrt{5}}{10} + \frac{1}{2}\right ) \log{\left (x + \frac{1}{2} + \frac{\sqrt{5}}{2} \right )} + \left (\frac{1}{2} - \frac{\sqrt{5}}{10}\right ) \log{\left (x - \frac{\sqrt{5}}{2} + \frac{1}{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04988, size = 54, normalized size = 1.1 \begin{align*} -\frac{1}{10} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - \sqrt{5} + 1 \right |}}{{\left | 2 \, x + \sqrt{5} + 1 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} + x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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