Optimal. Leaf size=21 \[ e^{-13+x-x^2} \log \left (\frac {x}{4+2 x}\right ) \]
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Rubi [B] time = 0.58, antiderivative size = 44, normalized size of antiderivative = 2.10, number of steps used = 4, number of rules used = 4, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {12, 1593, 6688, 2288} \begin {gather*} \frac {e^{-x^2+x-13} \left (-2 x^2-3 x+2\right ) \log \left (\frac {x}{2 (x+2)}\right )}{(1-2 x) (x+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1593
Rule 2288
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2 e^{-11+x-x^2}+e^{-11+x-x^2} \left (2 x-3 x^2-2 x^3\right ) \log \left (\frac {x}{4+2 x}\right )}{2 x+x^2} \, dx}{e^2}\\ &=\frac {\int \frac {2 e^{-11+x-x^2}+e^{-11+x-x^2} \left (2 x-3 x^2-2 x^3\right ) \log \left (\frac {x}{4+2 x}\right )}{x (2+x)} \, dx}{e^2}\\ &=\frac {\int \frac {e^{-11+x-x^2} \left (2+x \left (2-3 x-2 x^2\right ) \log \left (\frac {x}{4+2 x}\right )\right )}{x (2+x)} \, dx}{e^2}\\ &=\frac {e^{-13+x-x^2} \left (2-3 x-2 x^2\right ) \log \left (\frac {x}{2 (2+x)}\right )}{(1-2 x) (2+x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} e^{-13+x-x^2} \log \left (\frac {x}{4+2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.30, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} + x - 13\right )} \log \left (\frac {x}{2 \, {\left (x + 2\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 19, normalized size = 0.90 \begin {gather*} e^{\left (-x^{2} + x - 13\right )} \log \left (\frac {x}{2 \, {\left (x + 2\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 25, normalized size = 1.19
method | result | size |
norman | \(\ln \left (\frac {x}{2 x +4}\right ) {\mathrm e}^{-2} {\mathrm e}^{-x^{2}+x -11}\) | \(25\) |
risch | \(-\ln \left (2+x \right ) {\mathrm e}^{-x^{2}+x -13}+\ln \relax (x ) {\mathrm e}^{-x^{2}+x -13}-\frac {i \mathrm {csgn}\left (\frac {i x}{2+x}\right ) \mathrm {csgn}\left (\frac {i}{2+x}\right ) \mathrm {csgn}\left (i x \right ) \pi \,{\mathrm e}^{-x^{2}+x -13}}{2}+\frac {i \mathrm {csgn}\left (\frac {i x}{2+x}\right )^{2} \mathrm {csgn}\left (i x \right ) \pi \,{\mathrm e}^{-x^{2}+x -13}}{2}+\frac {i \mathrm {csgn}\left (\frac {i x}{2+x}\right )^{2} \mathrm {csgn}\left (\frac {i}{2+x}\right ) \pi \,{\mathrm e}^{-x^{2}+x -13}}{2}-\frac {i \mathrm {csgn}\left (\frac {i x}{2+x}\right )^{3} \pi \,{\mathrm e}^{-x^{2}+x -13}}{2}-\ln \relax (2) {\mathrm e}^{-x^{2}+x -13}\) | \(168\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 28, normalized size = 1.33 \begin {gather*} -{\left ({\left (\log \relax (2) - \log \relax (x)\right )} e^{x} + e^{x} \log \left (x + 2\right )\right )} e^{\left (-x^{2} - 13\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 21, normalized size = 1.00 \begin {gather*} {\mathrm {e}}^{-13}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^x\,\ln \left (\frac {x}{2\,x+4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 19, normalized size = 0.90 \begin {gather*} \frac {e^{- x^{2} + x - 11} \log {\left (\frac {x}{2 x + 4} \right )}}{e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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