Optimal. Leaf size=51 \[ \frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (-x-\sqrt{5}+2\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right ) \]
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Rubi [A] time = 0.0158961, antiderivative size = 51, normalized size of antiderivative = 1., 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 = {632, 31} \[ \frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (-x-\sqrt{5}+2\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right ) \]
Antiderivative was successfully verified.
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Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{2+x}{-1-4 x+x^2} \, dx &=-\left (\frac{1}{10} \left (-5+4 \sqrt{5}\right ) \int \frac{1}{-2+\sqrt{5}+x} \, dx\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \int \frac{1}{-2-\sqrt{5}+x} \, dx\\ &=\frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (2-\sqrt{5}-x\right )+\frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (2+\sqrt{5}-x\right )\\ \end{align*}
Mathematica [A] time = 0.0256704, size = 47, normalized size = 0.92 \[ \frac{1}{10} \left (5+4 \sqrt{5}\right ) \log \left (-x+\sqrt{5}+2\right )+\frac{1}{10} \left (5-4 \sqrt{5}\right ) \log \left (x+\sqrt{5}-2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 29, normalized size = 0.6 \begin{align*}{\frac{\ln \left ({x}^{2}-4\,x-1 \right ) }{2}}-{\frac{4\,\sqrt{5}}{5}{\it Artanh} \left ({\frac{ \left ( 2\,x-4 \right ) \sqrt{5}}{10}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48026, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{x - \sqrt{5} - 2}{x + \sqrt{5} - 2}\right ) + \frac{1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81323, size = 128, normalized size = 2.51 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{x^{2} - 2 \, \sqrt{5}{\left (x - 2\right )} - 4 \, x + 9}{x^{2} - 4 \, x - 1}\right ) + \frac{1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.104519, size = 42, normalized size = 0.82 \begin{align*} \left (\frac{1}{2} - \frac{2 \sqrt{5}}{5}\right ) \log{\left (x - 2 + \sqrt{5} \right )} + \left (\frac{1}{2} + \frac{2 \sqrt{5}}{5}\right ) \log{\left (x - \sqrt{5} - 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.04397, size = 59, normalized size = 1.16 \begin{align*} \frac{2}{5} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{5} - 4 \right |}}{{\left | 2 \, x + 2 \, \sqrt{5} - 4 \right |}}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} - 4 \, x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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