Integrand size = 39, antiderivative size = 24 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2-e^{-6-x-x^2} (3+x) \log (\log (5)) \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 15, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6874, 2266, 2236, 2272, 2273} \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2-e^{-x^2-x-6} x \log (\log (5))-3 e^{-x^2-x-6} \log (\log (5)) \]
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Rule 2236
Rule 2266
Rule 2272
Rule 2273
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (2 x+e^{-6-x-x^2} \left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx \\ & = x^2+\log (\log (5)) \int e^{-6-x-x^2} \left (2+7 x+2 x^2\right ) \, dx \\ & = x^2+\log (\log (5)) \int \left (2 e^{-6-x-x^2}+7 e^{-6-x-x^2} x+2 e^{-6-x-x^2} x^2\right ) \, dx \\ & = x^2+(2 \log (\log (5))) \int e^{-6-x-x^2} \, dx+(2 \log (\log (5))) \int e^{-6-x-x^2} x^2 \, dx+(7 \log (\log (5))) \int e^{-6-x-x^2} x \, dx \\ & = x^2-\frac {7}{2} e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))+\log (\log (5)) \int e^{-6-x-x^2} \, dx-\log (\log (5)) \int e^{-6-x-x^2} x \, dx-\frac {1}{2} (7 \log (\log (5))) \int e^{-6-x-x^2} \, dx+\frac {(2 \log (\log (5))) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{e^{23/4}} \\ & = x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))-\frac {\sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right ) \log (\log (5))}{e^{23/4}}+\frac {1}{2} \log (\log (5)) \int e^{-6-x-x^2} \, dx+\frac {\log (\log (5)) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{e^{23/4}}-\frac {(7 \log (\log (5))) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{2 e^{23/4}} \\ & = x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5))+\frac {\sqrt {\pi } \text {erf}\left (\frac {1}{2} (-1-2 x)\right ) \log (\log (5))}{4 e^{23/4}}+\frac {\log (\log (5)) \int e^{-\frac {1}{4} (-1-2 x)^2} \, dx}{2 e^{23/4}} \\ & = x^2-3 e^{-6-x-x^2} \log (\log (5))-e^{-6-x-x^2} x \log (\log (5)) \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2+e^{-x-x^2} \left (-\frac {3}{e^6}-\frac {x}{e^6}\right ) \log (\log (5)) \]
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Time = 0.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
method | result | size |
risch | \(x^{2}-\ln \left (\ln \left (5\right )\right ) \left (3+x \right ) {\mathrm e}^{-x^{2}-x -6}\) | \(24\) |
norman | \(\left (x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}-x \ln \left (\ln \left (5\right )\right )-3 \ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}\) | \(38\) |
parallelrisch | \(\left (x^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}+6}-x \ln \left (\ln \left (5\right )\right )-3 \ln \left (\ln \left (5\right )\right )\right ) {\mathrm e}^{-x} {\mathrm e}^{-x^{2}-6}\) | \(38\) |
parts | \(-3 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) {\mathrm e}^{-x^{2}-x}-{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) x \,{\mathrm e}^{-x^{2}-x}+x^{2}\) | \(44\) |
default | \(x^{2}+{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )+7 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \left (-\frac {{\mathrm e}^{-x^{2}-x}}{2}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )}{4}\right )+2 \,{\mathrm e}^{-6} \ln \left (\ln \left (5\right )\right ) \left (-\frac {x \,{\mathrm e}^{-x^{2}-x}}{2}+\frac {{\mathrm e}^{-x^{2}-x}}{4}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{\frac {1}{4}} \operatorname {erf}\left (\frac {1}{2}+x \right )}{8}\right )\) | \(101\) |
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Time = 0.40 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx={\left (x^{2} e^{\left (x^{2} + x + 6\right )} - {\left (x + 3\right )} \log \left (\log \left (5\right )\right )\right )} e^{\left (-x^{2} - x - 6\right )} \]
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Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^{2} + \left (- x \log {\left (\log {\left (5 \right )} \right )} - 3 \log {\left (\log {\left (5 \right )} \right )}\right ) e^{- x} e^{- x^{2} - 6} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 152, normalized size of antiderivative = 6.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=\sqrt {\pi } \operatorname {erf}\left (x + \frac {1}{2}\right ) e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) - \frac {1}{4} i \, {\left (-\frac {4 i \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{{\left ({\left (2 \, x + 1\right )}^{2}\right )}^{\frac {3}{2}}} + \frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} + 4 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) - \frac {7}{4} i \, {\left (-\frac {i \, \sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (2 \, x + 1\right )}^{2}}} - 2 i \, e^{\left (-\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {23}{4}\right )} \log \left (\log \left (5\right )\right ) + x^{2} \]
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Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^{2} - \frac {1}{2} \, {\left ({\left (2 \, x + 1\right )} \log \left (\log \left (5\right )\right ) + 5 \, \log \left (\log \left (5\right )\right )\right )} e^{\left (-x^{2} - x - 6\right )} \]
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Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int e^{-6-x-x^2} \left (2 e^{6+x+x^2} x+\left (2+7 x+2 x^2\right ) \log (\log (5))\right ) \, dx=x^2-3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \left (5\right )\right )-x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-6}\,{\mathrm {e}}^{-x^2}\,\ln \left (\ln \left (5\right )\right ) \]
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