Integrand size = 113, antiderivative size = 30 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {5}{4} e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \]
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\[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\int \frac {\exp \left (-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}\right ) \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int \frac {\exp \left (-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}\right ) \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx \\ & = \frac {1}{4} \int \frac {5 e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10 x+\left (-1+e^{\frac {5}{x^2}}\right ) x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx \\ & = \frac {5}{4} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10 x+\left (-1+e^{\frac {5}{x^2}}\right ) x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^3} \, dx \\ & = \frac {5}{4} \int \left (\frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10-x^2\right )}{x^2}+\frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (x^3-2 \log (x)+2 \log ^2(x)\right )}{x^3}\right ) \, dx \\ & = \frac {5}{4} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (-10-x^2\right )}{x^2} \, dx+\frac {5}{4} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \left (x^3-2 \log (x)+2 \log ^2(x)\right )}{x^3} \, dx \\ & = \frac {5}{4} \int \left (-e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}-\frac {10 e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}}{x^2}\right ) \, dx+\frac {5}{4} \int \left (e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}-\frac {2 e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log (x)}{x^3}+\frac {2 e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log ^2(x)}{x^3}\right ) \, dx \\ & = \frac {5}{4} \int e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \, dx-\frac {5}{4} \int e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \, dx-\frac {5}{2} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log (x)}{x^3} \, dx+\frac {5}{2} \int \frac {e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \log ^2(x)}{x^3} \, dx-\frac {25}{2} \int \frac {e^{-e-\frac {5}{x^2}+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}}}{x^2} \, dx \\ \end{align*}
Time = 2.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {5}{4} e^{-e+x-e^{-\frac {5}{x^2}} x-\frac {\log ^2(x)}{x^2}} \]
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Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {5 \,{\mathrm e}^{-\frac {{\mathrm e}^{-\frac {5}{x^{2}}} x^{3}+x^{2} {\mathrm e}-x^{3}+\ln \left (x \right )^{2}}{x^{2}}}}{4}\) | \(35\) |
derivativedivides | \(\frac {{\mathrm e}^{\frac {\left (-{\mathrm e}^{\frac {5}{x^{2}}} \ln \left (x \right )^{2}+\left (x^{2} \ln \left (5\right )-x^{2} {\mathrm e}+x^{3}\right ) {\mathrm e}^{\frac {5}{x^{2}}}-x^{3}\right ) {\mathrm e}^{-\frac {5}{x^{2}}}}{x^{2}}}}{4}\) | \(58\) |
default | \(\frac {{\mathrm e}^{\frac {\left (-{\mathrm e}^{\frac {5}{x^{2}}} \ln \left (x \right )^{2}+\left (x^{2} \ln \left (5\right )-x^{2} {\mathrm e}+x^{3}\right ) {\mathrm e}^{\frac {5}{x^{2}}}-x^{3}\right ) {\mathrm e}^{-\frac {5}{x^{2}}}}{x^{2}}}}{4}\) | \(58\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {\left (-{\mathrm e}^{\frac {5}{x^{2}}} \ln \left (x \right )^{2}+\left (x^{2} \ln \left (5\right )-x^{2} {\mathrm e}+x^{3}\right ) {\mathrm e}^{\frac {5}{x^{2}}}-x^{3}\right ) {\mathrm e}^{-\frac {5}{x^{2}}}}{x^{2}}}}{4}\) | \(58\) |
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {1}{4} \, e^{\left (-\frac {{\left (x^{3} + e^{\left (\frac {5}{x^{2}}\right )} \log \left (x\right )^{2} - {\left (x^{3} - x^{2} e + x^{2} \log \left (5\right ) - 5\right )} e^{\left (\frac {5}{x^{2}}\right )}\right )} e^{\left (-\frac {5}{x^{2}}\right )}}{x^{2}} + \frac {5}{x^{2}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {e^{\frac {\left (- x^{3} + \left (x^{3} - e x^{2} + x^{2} \log {\left (5 \right )}\right ) e^{\frac {5}{x^{2}}} - e^{\frac {5}{x^{2}}} \log {\left (x \right )}^{2}\right ) e^{- \frac {5}{x^{2}}}}{x^{2}}}}{4} \]
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Time = 0.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {5}{4} \, e^{\left (-x e^{\left (-\frac {5}{x^{2}}\right )} + x - \frac {\log \left (x\right )^{2}}{x^{2}} - e\right )} \]
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\[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\int { \frac {{\left (x^{3} e^{\left (\frac {5}{x^{2}}\right )} - x^{3} + 2 \, e^{\left (\frac {5}{x^{2}}\right )} \log \left (x\right )^{2} - 2 \, e^{\left (\frac {5}{x^{2}}\right )} \log \left (x\right ) - 10 \, x\right )} e^{\left (-\frac {{\left (x^{3} + e^{\left (\frac {5}{x^{2}}\right )} \log \left (x\right )^{2} - {\left (x^{3} - x^{2} e + x^{2} \log \left (5\right )\right )} e^{\left (\frac {5}{x^{2}}\right )}\right )} e^{\left (-\frac {5}{x^{2}}\right )}}{x^{2}} - \frac {5}{x^{2}}\right )}}{4 \, x^{3}} \,d x } \]
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Time = 8.39 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {5}{x^2}+\frac {e^{-\frac {5}{x^2}} \left (-x^3+e^{\frac {5}{x^2}} \left (-e x^2+x^3+x^2 \log (5)\right )-e^{\frac {5}{x^2}} \log ^2(x)\right )}{x^2}} \left (-10 x-x^3+e^{\frac {5}{x^2}} x^3-2 e^{\frac {5}{x^2}} \log (x)+2 e^{\frac {5}{x^2}} \log ^2(x)\right )}{4 x^3} \, dx=\frac {5\,{\mathrm {e}}^{-\mathrm {e}}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-\frac {5}{x^2}}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-\frac {{\ln \left (x\right )}^2}{x^2}}}{4} \]
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