Optimal. Leaf size=30 \[ \frac {-2+x-\frac {x}{-6+3 e^{x-x^2} \log (x)}}{3 x} \]
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Rubi [F]
time = 4.30, 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 {24 e^{-2 x+2 x^2}+e^{-x+x^2} x+e^{-x+x^2} \left (-24+x^2-2 x^3\right ) \log (x)+6 \log ^2(x)}{36 e^{-2 x+2 x^2} x^2-36 e^{-x+x^2} x^2 \log (x)+9 x^2 \log ^2(x)} \, 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 (24 e^{-2 x+2 x^2}+e^{-x+x^2} x+e^{-x+x^2} \left (-24+x^2-2 x^3\right ) \log (x)+6 \log ^2(x)\right )}{9 x^2 \left (2 e^{x^2}-e^x \log (x)\right )^2} \, dx\\ &=\frac {1}{9} \int \frac {e^{2 x} \left (24 e^{-2 x+2 x^2}+e^{-x+x^2} x+e^{-x+x^2} \left (-24+x^2-2 x^3\right ) \log (x)+6 \log ^2(x)\right )}{x^2 \left (2 e^{x^2}-e^x \log (x)\right )^2} \, dx\\ &=\frac {1}{9} \int \left (\frac {6}{x^2}-\frac {e^{2 x} \log (x) \left (-1-x \log (x)+2 x^2 \log (x)\right )}{2 x \left (-2 e^{x^2}+e^x \log (x)\right )^2}+\frac {e^x \left (-1-x \log (x)+2 x^2 \log (x)\right )}{2 x \left (-2 e^{x^2}+e^x \log (x)\right )}\right ) \, dx\\ &=-\frac {2}{3 x}-\frac {1}{18} \int \frac {e^{2 x} \log (x) \left (-1-x \log (x)+2 x^2 \log (x)\right )}{x \left (-2 e^{x^2}+e^x \log (x)\right )^2} \, dx+\frac {1}{18} \int \frac {e^x \left (-1-x \log (x)+2 x^2 \log (x)\right )}{x \left (-2 e^{x^2}+e^x \log (x)\right )} \, dx\\ &=-\frac {2}{3 x}-\frac {1}{18} \int \left (-\frac {e^{2 x} \log (x)}{x \left (-2 e^{x^2}+e^x \log (x)\right )^2}-\frac {e^{2 x} \log ^2(x)}{\left (-2 e^{x^2}+e^x \log (x)\right )^2}+\frac {2 e^{2 x} x \log ^2(x)}{\left (-2 e^{x^2}+e^x \log (x)\right )^2}\right ) \, dx+\frac {1}{18} \int \left (\frac {e^x}{x \left (2 e^{x^2}-e^x \log (x)\right )}-\frac {e^x \log (x)}{-2 e^{x^2}+e^x \log (x)}+\frac {2 e^x x \log (x)}{-2 e^{x^2}+e^x \log (x)}\right ) \, dx\\ &=-\frac {2}{3 x}+\frac {1}{18} \int \frac {e^x}{x \left (2 e^{x^2}-e^x \log (x)\right )} \, dx+\frac {1}{18} \int \frac {e^{2 x} \log (x)}{x \left (-2 e^{x^2}+e^x \log (x)\right )^2} \, dx+\frac {1}{18} \int \frac {e^{2 x} \log ^2(x)}{\left (-2 e^{x^2}+e^x \log (x)\right )^2} \, dx-\frac {1}{18} \int \frac {e^x \log (x)}{-2 e^{x^2}+e^x \log (x)} \, dx-\frac {1}{9} \int \frac {e^{2 x} x \log ^2(x)}{\left (-2 e^{x^2}+e^x \log (x)\right )^2} \, dx+\frac {1}{9} \int \frac {e^x x \log (x)}{-2 e^{x^2}+e^x \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.24, size = 33, normalized size = 1.10 \begin {gather*} \frac {1}{9} \left (-\frac {6}{x}-\frac {e^{x^2}}{-2 e^{x^2}+e^x \log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 30, normalized size = 1.00
method | result | size |
risch | \(-\frac {2}{3 x}+\frac {{\mathrm e}^{x \left (x -1\right )}}{18 \,{\mathrm e}^{x \left (x -1\right )}-9 \ln \left (x \right )}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.39, size = 33, normalized size = 1.10 \begin {gather*} -\frac {{\left (x + 12\right )} e^{x} \log \left (x\right ) - 24 \, e^{\left (x^{2}\right )}}{18 \, {\left (x e^{x} \log \left (x\right ) - 2 \, x e^{\left (x^{2}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 38, normalized size = 1.27 \begin {gather*} \frac {{\left (x - 12\right )} e^{\left (x^{2} - x\right )} + 6 \, \log \left (x\right )}{9 \, {\left (2 \, x e^{\left (x^{2} - x\right )} - x \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 20, normalized size = 0.67 \begin {gather*} \frac {\log {\left (x \right )}}{36 e^{x^{2} - x} - 18 \log {\left (x \right )}} - \frac {2}{3 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 40, normalized size = 1.33 \begin {gather*} \frac {x \log \left (x\right ) - 24 \, e^{\left (x^{2} - x\right )} + 12 \, \log \left (x\right )}{18 \, {\left (2 \, x e^{\left (x^{2} - x\right )} - x \log \left (x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.95, size = 26, normalized size = 0.87 \begin {gather*} \frac {\ln \left (x\right )}{18\,\left (2\,{\mathrm {e}}^{x^2-x}-\ln \left (x\right )\right )}-\frac {2}{3\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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