Integrand size = 55, antiderivative size = 21 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=\frac {2 x \left (11-e^{-x^2} x^x\right )}{\log (x)} \]
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Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {6820, 6874, 2407, 2334, 2335, 2326} \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=\frac {22 x}{\log (x)}-\frac {2 e^{-x^2} x^{x+1}}{\log (x)} \]
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Rule 2326
Rule 2334
Rule 2335
Rule 2407
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-22+22 \log (x)-2 e^{-x^2} x^x \left (-1+\left (1+x-2 x^2\right ) \log (x)+x \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = \int \left (\frac {22 (-1+\log (x))}{\log ^2(x)}-\frac {2 e^{-x^2} x^x \left (-1+\log (x)+x \log (x)-2 x^2 \log (x)+x \log ^2(x)\right )}{\log ^2(x)}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-x^2} x^x \left (-1+\log (x)+x \log (x)-2 x^2 \log (x)+x \log ^2(x)\right )}{\log ^2(x)} \, dx\right )+22 \int \frac {-1+\log (x)}{\log ^2(x)} \, dx \\ & = -\frac {2 e^{-x^2} x^{1+x}}{\log (x)}+22 \int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx \\ & = -\frac {2 e^{-x^2} x^{1+x}}{\log (x)}-22 \int \frac {1}{\log ^2(x)} \, dx+22 \int \frac {1}{\log (x)} \, dx \\ & = \frac {22 x}{\log (x)}-\frac {2 e^{-x^2} x^{1+x}}{\log (x)}+22 \operatorname {LogIntegral}(x)-22 \int \frac {1}{\log (x)} \, dx \\ & = \frac {22 x}{\log (x)}-\frac {2 e^{-x^2} x^{1+x}}{\log (x)} \\ \end{align*}
Time = 3.55 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=\frac {2 x \left (11-e^{-x^2} x^x\right )}{\log (x)} \]
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Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19
method | result | size |
risch | \(\frac {22 x}{\ln \left (x \right )}-\frac {2 x \,{\mathrm e}^{-x^{2}} x^{x}}{\ln \left (x \right )}\) | \(25\) |
parallelrisch | \(\frac {\left (22 \,{\mathrm e}^{x^{2}} x -2 \,{\mathrm e}^{x \ln \left (x \right )} x \right ) {\mathrm e}^{-x^{2}}}{\ln \left (x \right )}\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=-\frac {2 \, {\left (x x^{x} - 11 \, x e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{\log \left (x\right )} \]
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Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=\frac {22 x}{\log {\left (x \right )}} - \frac {2 x e^{- x^{2}} e^{x \log {\left (x \right )}}}{\log {\left (x \right )}} \]
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Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=-\frac {2 \, {\left (x x^{x} - 11 \, x e^{\left (x^{2}\right )}\right )} e^{\left (-x^{2}\right )}}{\log \left (x\right )} \]
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Time = 0.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=-\frac {2 \, x e^{\left (-x^{2} + x \log \left (x\right )\right )}}{\log \left (x\right )} + \frac {22 \, x}{\log \left (x\right )} \]
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Time = 13.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{-x^2} \left (-22 e^{x^2}+22 e^{x^2} \log (x)+x^x \left (2+\left (-2-2 x+4 x^2\right ) \log (x)-2 x \log ^2(x)\right )\right )}{\log ^2(x)} \, dx=\frac {2\,x\,{\mathrm {e}}^{-x^2}\,\left (11\,{\mathrm {e}}^{x^2}-x^x\right )}{\ln \left (x\right )} \]
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