Optimal. Leaf size=17 \[ \frac {\log (3)}{x \left (e^{-2-x}+x\right )} \]
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Rubi [F] time = 1.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-2-x} (-1+x) \log (3)-2 x \log (3)}{e^{-4-2 x} x^2+2 e^{-2-x} x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2+x} \left (-1-\left (-1+2 e^{2+x}\right ) x\right ) \log (3)}{x^2 \left (1+e^{2+x} x\right )^2} \, dx\\ &=\log (3) \int \frac {e^{2+x} \left (-1-\left (-1+2 e^{2+x}\right ) x\right )}{x^2 \left (1+e^{2+x} x\right )^2} \, dx\\ &=\log (3) \int \left (\frac {e^{2+x} (1+x)}{x^2 \left (1+e^{2+x} x\right )^2}-\frac {2 e^{2+x}}{x^2 \left (1+e^{2+x} x\right )}\right ) \, dx\\ &=\log (3) \int \frac {e^{2+x} (1+x)}{x^2 \left (1+e^{2+x} x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{2+x}}{x^2 \left (1+e^{2+x} x\right )} \, dx\\ &=\log (3) \int \left (\frac {e^{2+x}}{x^2 \left (1+e^{2+x} x\right )^2}+\frac {e^{2+x}}{x \left (1+e^{2+x} x\right )^2}\right ) \, dx-(2 \log (3)) \int \frac {e^{2+x}}{x^2 \left (1+e^{2+x} x\right )} \, dx\\ &=\log (3) \int \frac {e^{2+x}}{x^2 \left (1+e^{2+x} x\right )^2} \, dx+\log (3) \int \frac {e^{2+x}}{x \left (1+e^{2+x} x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{2+x}}{x^2 \left (1+e^{2+x} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 25, normalized size = 1.47 \begin {gather*} -\left (\left (-\frac {1}{x^2}+\frac {1}{x^2 \left (1+e^{2+x} x\right )}\right ) \log (3)\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log \relax (3)}{x^{2} + x e^{\left (-x - 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 19, normalized size = 1.12 \begin {gather*} \frac {e^{\left (x + 2\right )} \log \relax (3)}{x^{2} e^{\left (x + 2\right )} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 17, normalized size = 1.00
method | result | size |
norman | \(\frac {\ln \relax (3)}{x \left (x +{\mathrm e}^{-x -2}\right )}\) | \(17\) |
risch | \(\frac {\ln \relax (3)}{x \left (x +{\mathrm e}^{-x -2}\right )}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 19, normalized size = 1.12 \begin {gather*} \frac {e^{\left (x + 2\right )} \log \relax (3)}{x^{2} e^{\left (x + 2\right )} + x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.16, size = 17, normalized size = 1.00 \begin {gather*} \frac {\ln \relax (3)}{x\,{\mathrm {e}}^{-x-2}+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} \frac {\log {\relax (3 )}}{x^{2} + x e^{- x - 2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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