Optimal. Leaf size=63 \[ -\frac {2 \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}{1-\sqrt {5}}-\frac {2 \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}{1+\sqrt {5}} \]
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Rubi [A] time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2086, 628} \[ -\frac {2 \log \left (2 x^2-\left (1-\sqrt {5}\right ) x+2\right )}{1-\sqrt {5}}-\frac {2 \log \left (2 x^2-\left (1+\sqrt {5}\right ) x+2\right )}{1+\sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 628
Rule 2086
Rubi steps
\begin {align*} \int \frac {2+x-4 x^2+2 x^3}{1-x+x^2-x^3+x^4} \, dx &=-\frac {\int \frac {-2 \sqrt {5}+\left (10-2 \sqrt {5}\right ) x}{2+\left (-1-\sqrt {5}\right ) x+2 x^2} \, dx}{\sqrt {5}}+\frac {\int \frac {2 \sqrt {5}+\left (10+2 \sqrt {5}\right ) x}{2+\left (-1+\sqrt {5}\right ) x+2 x^2} \, dx}{\sqrt {5}}\\ &=-\frac {2 \log \left (2-\left (1-\sqrt {5}\right ) x+2 x^2\right )}{1-\sqrt {5}}-\frac {2 \log \left (2-\left (1+\sqrt {5}\right ) x+2 x^2\right )}{1+\sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.87 \[ \frac {1}{2} \left (\left (1+\sqrt {5}\right ) \log \left (2 x^2+\left (\sqrt {5}-1\right ) x+2\right )-\left (\sqrt {5}-1\right ) \log \left (-2 x^2+\sqrt {5} x+x-2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 83, normalized size = 1.32 \[ \frac {1}{2} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - 2 \, x^{3} + 7 \, x^{2} + \sqrt {5} {\left (2 \, x^{3} - x^{2} + 2 \, x\right )} - 2 \, x + 2}{x^{4} - x^{3} + x^{2} - x + 1}\right ) + \frac {1}{2} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 58, normalized size = 0.92 \[ -\frac {1}{2} \, \sqrt {5} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} + 1\right )} + 1\right ) + \frac {1}{2} \, \sqrt {5} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} - 1\right )} + 1\right ) + \frac {1}{2} \, \log \left (x^{4} - x^{3} + x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 82, normalized size = 1.30 \[ \frac {\ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{2}-\frac {\sqrt {5}\, \ln \left (2 x^{2}-\sqrt {5}\, x -x +2\right )}{2}+\frac {\ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{2}+\frac {\sqrt {5}\, \ln \left (2 x^{2}+\sqrt {5}\, x -x +2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2 \, x^{3} - 4 \, x^{2} + x + 2}{x^{4} - x^{3} + x^{2} - x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 75, normalized size = 1.19 \[ \frac {\ln \left (x^2-\frac {\sqrt {5}\,x}{2}-\frac {x}{2}+1\right )}{2}+\frac {\ln \left (\frac {\sqrt {5}\,x}{2}-\frac {x}{2}+x^2+1\right )}{2}-\frac {\sqrt {5}\,\ln \left (x^2-\frac {\sqrt {5}\,x}{2}-\frac {x}{2}+1\right )}{2}+\frac {\sqrt {5}\,\ln \left (\frac {\sqrt {5}\,x}{2}-\frac {x}{2}+x^2+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 58, normalized size = 0.92 \[ \left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right ) \log {\left (x^{2} + x \left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right ) + 1 \right )} + \left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right ) \log {\left (x^{2} + x \left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right ) + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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