Optimal. Leaf size=24 \[ e^{\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}}-x \]
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Rubi [F] time = 2.06, 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^{-\frac {x^2}{1+x}} \left (e^{\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} \left (-4 x-2 x^2\right )+e^{\frac {x^2}{1+x}} \left (-1-2 x-x^2\right ) \log (4)\right )}{\left (1+2 x+x^2\right ) \log (4)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-\frac {x^2}{1+x}} \left (e^{\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} \left (-4 x-2 x^2\right )+e^{\frac {x^2}{1+x}} \left (-1-2 x-x^2\right ) \log (4)\right )}{1+2 x+x^2} \, dx}{\log (4)}\\ &=\frac {\int \frac {e^{-\frac {x^2}{1+x}} \left (e^{\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} \left (-4 x-2 x^2\right )+e^{\frac {x^2}{1+x}} \left (-1-2 x-x^2\right ) \log (4)\right )}{(1+x)^2} \, dx}{\log (4)}\\ &=\frac {\int \left (-\frac {2 e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} x (2+x)}{(1+x)^2}-\log (4)\right ) \, dx}{\log (4)}\\ &=-x-\frac {2 \int \frac {e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} x (2+x)}{(1+x)^2} \, dx}{\log (4)}\\ &=-x-\frac {2 \int \left (e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}}-\frac {e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}}}{(1+x)^2}\right ) \, dx}{\log (4)}\\ &=-x-\frac {2 \int e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}} \, dx}{\log (4)}+\frac {2 \int \frac {e^{-\frac {x^2}{1+x}+\frac {2 e^{-\frac {x^2}{1+x}}}{\log (4)}}}{(1+x)^2} \, dx}{\log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {2 e^{1-x-\frac {1}{1+x}}}{\log (4)}}-x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 21, normalized size = 0.88 \begin {gather*} -x + e^{\left (\frac {e^{\left (-\frac {x^{2}}{x + 1}\right )}}{\log \relax (2)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left ({\left (x^{2} + 2 \, x + 1\right )} e^{\left (\frac {x^{2}}{x + 1}\right )} \log \relax (2) + {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {e^{\left (-\frac {x^{2}}{x + 1}\right )}}{\log \relax (2)}\right )}\right )} e^{\left (-\frac {x^{2}}{x + 1}\right )}}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (2)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 22, normalized size = 0.92
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}}{x +1}}}{\ln \relax (2)}}-x\) | \(22\) |
norman | \(\frac {\left ({\mathrm e}^{\frac {x^{2}}{x +1}}+{\mathrm e}^{\frac {x^{2}}{x +1}} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}}{x +1}}}{\ln \relax (2)}}+x \,{\mathrm e}^{\frac {x^{2}}{x +1}} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {x^{2}}{x +1}}}{\ln \relax (2)}}-x^{2} {\mathrm e}^{\frac {x^{2}}{x +1}}\right ) {\mathrm e}^{-\frac {x^{2}}{x +1}}}{x +1}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.06, size = 75, normalized size = 3.12 \begin {gather*} -\frac {{\left (\frac {x^{2} + x - 1}{x + 1} - 2 \, \log \left (x + 1\right )\right )} \log \relax (2) + 2 \, {\left (\frac {1}{x + 1} + \log \left (x + 1\right )\right )} \log \relax (2) - e^{\left (\frac {e^{\left (-x - \frac {1}{x + 1} + 1\right )}}{\log \relax (2)}\right )} \log \relax (2) - \frac {\log \relax (2)}{x + 1}}{\log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 21, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^{-\frac {x^2}{x+1}}}{\ln \relax (2)}}-x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 15, normalized size = 0.62 \begin {gather*} - x + e^{\frac {e^{- \frac {x^{2}}{x + 1}}}{\log {\relax (2 )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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