Optimal. Leaf size=45 \[ \frac {1}{4} \left (2+\sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )+\frac {1}{4} \left (2-\sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 45, 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} \[ \frac {1}{4} \left (2+\sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )+\frac {1}{4} \left (2-\sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 632
Rubi steps
\begin {align*} \int \frac {2+x}{-1+2 x+x^2} \, dx &=-\left (\frac {1}{4} \left (-2+\sqrt {2}\right ) \int \frac {1}{1+\sqrt {2}+x} \, dx\right )+\frac {1}{4} \left (2+\sqrt {2}\right ) \int \frac {1}{1-\sqrt {2}+x} \, dx\\ &=\frac {1}{4} \left (2+\sqrt {2}\right ) \log \left (1-\sqrt {2}+x\right )+\frac {1}{4} \left (2-\sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.93 \[ \frac {1}{4} \left (\left (2+\sqrt {2}\right ) \log \left (-x+\sqrt {2}-1\right )-\left (\sqrt {2}-2\right ) \log \left (x+\sqrt {2}+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 45, normalized size = 1.00 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 44, normalized size = 0.98 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + 2 \right |}}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.64 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{2}+\frac {\ln \left (x^{2}+2 x -1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.62, size = 35, normalized size = 0.78 \[ \frac {1}{4} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} + 1}{x + \sqrt {2} + 1}\right ) + \frac {1}{2} \, \log \left (x^{2} + 2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 34, normalized size = 0.76 \[ \ln \left (x-\sqrt {2}+1\right )\,\left (\frac {\sqrt {2}}{4}+\frac {1}{2}\right )-\ln \left (x+\sqrt {2}+1\right )\,\left (\frac {\sqrt {2}}{4}-\frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 39, normalized size = 0.87 \[ \left (\frac {1}{2} - \frac {\sqrt {2}}{4}\right ) \log {\left (x + 1 + \sqrt {2} \right )} + \left (\frac {\sqrt {2}}{4} + \frac {1}{2}\right ) \log {\left (x - \sqrt {2} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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