Optimal. Leaf size=19 \[ e^2-\frac {x^2}{8 e^4 (-1+x)} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.21, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 27, 683} \begin {gather*} \frac {1}{8 e^4 (1-x)}-\frac {x}{8 e^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 683
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2 x-x^2}{4-8 x+4 x^2} \, dx}{2 e^4}\\ &=\frac {\int \frac {2 x-x^2}{4 (-1+x)^2} \, dx}{2 e^4}\\ &=\frac {\int \frac {2 x-x^2}{(-1+x)^2} \, dx}{8 e^4}\\ &=\frac {\int \left (-1+\frac {1}{(-1+x)^2}\right ) \, dx}{8 e^4}\\ &=\frac {1}{8 e^4 (1-x)}-\frac {x}{8 e^4}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 0.74 \begin {gather*} -\frac {\frac {1}{-1+x}+x}{8 e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 22, normalized size = 1.16 \begin {gather*} -\frac {{\left (x^{2} - x + 1\right )} e^{\left (-\log \relax (2) - 4\right )}}{4 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 16, normalized size = 0.84 \begin {gather*} -\frac {1}{4} \, {\left (x + \frac {1}{x - 1}\right )} e^{\left (-\log \relax (2) - 4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 15, normalized size = 0.79
method | result | size |
norman | \(-\frac {x^{2} {\mathrm e}^{-4}}{8 \left (x -1\right )}\) | \(15\) |
risch | \(-\frac {{\mathrm e}^{-4} x}{8}-\frac {{\mathrm e}^{-4}}{8 \left (x -1\right )}\) | \(16\) |
gosper | \(-\frac {x^{2} {\mathrm e}^{-4}}{8 \left (x -1\right )}\) | \(18\) |
default | \(\frac {{\mathrm e}^{-4} \left (-x -\frac {1}{x -1}\right )}{8}\) | \(21\) |
meijerg | \(\frac {{\mathrm e}^{-4-\ln \relax (2)} \left (-\frac {x \left (-3 x +6\right )}{3 \left (1-x \right )}-2 \ln \left (1-x \right )\right )}{4}+\frac {{\mathrm e}^{-4-\ln \relax (2)} \left (\frac {x}{1-x}+\ln \left (1-x \right )\right )}{2}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 11, normalized size = 0.58 \begin {gather*} -\frac {1}{8} \, {\left (x + \frac {1}{x - 1}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.36, size = 17, normalized size = 0.89 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-4}}{8}-\frac {{\mathrm {e}}^{-4}}{8\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 1.05 \begin {gather*} - \frac {x}{8 e^{4}} - \frac {1}{8 x e^{4} - 8 e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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