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 = 0.02, antiderivative size = 51, 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 = {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 31
Rule 632
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*}
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Mathematica [A] time = 0.03, 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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IntegrateAlgebraic [A] time = 0.04, 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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fricas [A] time = 0.84, size = 45, normalized size = 0.88 \[ \frac {2}{5} \, \sqrt {5} \log \left (\frac {x^{2} - 2 \, \sqrt {5} {\left (x - 2\right )} - 4 \, x + 9}{x^{2} - 4 \, x - 1}\right ) + \frac {1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 44, normalized size = 0.86 \[ \frac {2}{5} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {5} - 4 \right |}}{{\left | 2 \, x + 2 \, \sqrt {5} - 4 \right |}}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} - 4 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 29, normalized size = 0.57
method | result | size |
default | \(\frac {\ln \left (x^{2}-4 x -1\right )}{2}-\frac {4 \sqrt {5}\, \arctanh \left (\frac {\left (2 x -4\right ) \sqrt {5}}{10}\right )}{5}\) | \(29\) |
risch | \(\frac {\ln \left (x -\sqrt {5}-2\right )}{2}+\frac {2 \ln \left (x -\sqrt {5}-2\right ) \sqrt {5}}{5}+\frac {\ln \left (x -2+\sqrt {5}\right )}{2}-\frac {2 \ln \left (x -2+\sqrt {5}\right ) \sqrt {5}}{5}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 35, normalized size = 0.69 \[ \frac {2}{5} \, \sqrt {5} \log \left (\frac {x - \sqrt {5} - 2}{x + \sqrt {5} - 2}\right ) + \frac {1}{2} \, \log \left (x^{2} - 4 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 34, normalized size = 0.67 \[ \ln \left (x-\sqrt {5}-2\right )\,\left (\frac {2\,\sqrt {5}}{5}+\frac {1}{2}\right )-\ln \left (x+\sqrt {5}-2\right )\,\left (\frac {2\,\sqrt {5}}{5}-\frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 42, normalized size = 0.82 \[ \left (\frac {1}{2} - \frac {2 \sqrt {5}}{5}\right ) \log {\left (x - 2 + \sqrt {5} \right )} + \left (\frac {1}{2} + \frac {2 \sqrt {5}}{5}\right ) \log {\left (x - \sqrt {5} - 2 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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