Integrand size = 150, antiderivative size = 31 \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=e^{x^2}+3 e^{-x^2 \left (4-x-\frac {x^4}{\log ^2(2)}\right )^2} \]
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Time = 10.70 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {12, 6873, 6874, 2257, 2240, 6838} \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=3 \exp \left (-\frac {x^2 \left (x^4+x \log ^2(2)-4 \log ^2(2)\right )^2}{\log ^4(2)}\right )+e^{x^2} \]
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Rule 12
Rule 2240
Rule 2257
Rule 6838
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \exp \left (-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}\right ) \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 \exp \left (x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}\right ) x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right ) \, dx}{\log ^4(2)} \\ & = \frac {\int 2 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (-15 x^8+72 x^4 \log ^2(2)-21 x^5 \log ^2(2)-48 \log ^4(2)+\exp \left (x^2 \left (17-8 x+x^2+\frac {x^8}{\log ^4(2)}-\frac {8 x^4}{\log ^2(2)}+\frac {2 x^5}{\log ^2(2)}\right )\right ) \log ^4(2)+36 x \log ^4(2)-6 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)} \\ & = \frac {2 \int \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (-15 x^8+72 x^4 \log ^2(2)-21 x^5 \log ^2(2)-48 \log ^4(2)+\exp \left (x^2 \left (17-8 x+x^2+\frac {x^8}{\log ^4(2)}-\frac {8 x^4}{\log ^2(2)}+\frac {2 x^5}{\log ^2(2)}\right )\right ) \log ^4(2)+36 x \log ^4(2)-6 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)} \\ & = \frac {2 \int \left (\exp \left (17 x^2-8 x^3+x^4+\frac {x^{10}}{\log ^4(2)}-\frac {8 x^6}{\log ^2(2)}+\frac {2 x^7}{\log ^2(2)}-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \log ^4(2)-3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (5 x^8-24 x^4 \log ^2(2)+7 x^5 \log ^2(2)+16 \log ^4(2)-12 x \log ^4(2)+2 x^2 \log ^4(2)\right )\right ) \, dx}{\log ^4(2)} \\ & = 2 \int \exp \left (17 x^2-8 x^3+x^4+\frac {x^{10}}{\log ^4(2)}-\frac {8 x^6}{\log ^2(2)}+\frac {2 x^7}{\log ^2(2)}-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \, dx-\frac {6 \int \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) x \left (5 x^8-24 x^4 \log ^2(2)+7 x^5 \log ^2(2)+16 \log ^4(2)-12 x \log ^4(2)+2 x^2 \log ^4(2)\right ) \, dx}{\log ^4(2)} \\ & = 3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right )+2 \int e^{x^2} x \, dx \\ & = e^{x^2}+3 \exp \left (-\frac {x^2 \left (x^4-4 \log ^2(2)+x \log ^2(2)\right )^2}{\log ^4(2)}\right ) \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=e^{-16 x^2} \left (e^{17 x^2}+3 e^{x^3 \left (8-x-\frac {x^7}{\log ^4(2)}+\frac {8 x^3}{\log ^2(2)}-\frac {2 x^4}{\log ^2(2)}\right )}\right ) \]
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Time = 2.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16
method | result | size |
risch | \({\mathrm e}^{x^{2}}+3 \,{\mathrm e}^{-\frac {x^{2} \left (x^{4}+x \ln \left (2\right )^{2}-4 \ln \left (2\right )^{2}\right )^{2}}{\ln \left (2\right )^{4}}}\) | \(36\) |
parts | \({\mathrm e}^{x^{2}}+3 \,{\mathrm e}^{-\frac {\left (x^{4}-8 x^{3}+16 x^{2}\right ) \ln \left (2\right )^{4}+\left (2 x^{7}-8 x^{6}\right ) \ln \left (2\right )^{2}+x^{10}}{\ln \left (2\right )^{4}}}\) | \(55\) |
parallelrisch | \(\frac {\left (3 \ln \left (2\right )^{4}+\ln \left (2\right )^{4} {\mathrm e}^{x^{2}} {\mathrm e}^{\frac {\left (x^{4}-8 x^{3}+16 x^{2}\right ) \ln \left (2\right )^{4}+\left (2 x^{7}-8 x^{6}\right ) \ln \left (2\right )^{2}+x^{10}}{\ln \left (2\right )^{4}}}\right ) {\mathrm e}^{-\frac {\left (x^{4}-8 x^{3}+16 x^{2}\right ) \ln \left (2\right )^{4}+\left (2 x^{7}-8 x^{6}\right ) \ln \left (2\right )^{2}+x^{10}}{\ln \left (2\right )^{4}}}}{\ln \left (2\right )^{4}}\) | \(114\) |
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (26) = 52\).
Time = 0.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 4.42 \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=e^{\left (\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 17 \, x^{2}\right )} \log \left (2\right )^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \left (2\right )^{2}}{\log \left (2\right )^{4}} - \frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \left (2\right )^{2}}{\log \left (2\right )^{4}}\right )} + 3 \, e^{\left (-\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \left (2\right )^{2}}{\log \left (2\right )^{4}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.58 \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=e^{x^{2}} + 3 e^{- \frac {x^{10} + \left (2 x^{7} - 8 x^{6}\right ) \log {\left (2 \right )}^{2} + \left (x^{4} - 8 x^{3} + 16 x^{2}\right ) \log {\left (2 \right )}^{4}}{\log {\left (2 \right )}^{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.46 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=\frac {3 \, e^{\left (-\frac {x^{10}}{\log \left (2\right )^{4}} - \frac {2 \, x^{7}}{\log \left (2\right )^{2}} - x^{4} + \frac {8 \, x^{6}}{\log \left (2\right )^{2}} + 8 \, x^{3} - 16 \, x^{2}\right )} \log \left (2\right )^{4} + e^{\left (x^{2}\right )} \log \left (2\right )^{4}}{\log \left (2\right )^{4}} \]
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\[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=\int { -\frac {2 \, {\left (15 \, x^{9} - x e^{\left (x^{2} + \frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \left (2\right )^{2}}{\log \left (2\right )^{4}}\right )} \log \left (2\right )^{4} + 6 \, {\left (x^{3} - 6 \, x^{2} + 8 \, x\right )} \log \left (2\right )^{4} + 3 \, {\left (7 \, x^{6} - 24 \, x^{5}\right )} \log \left (2\right )^{2}\right )} e^{\left (-\frac {x^{10} + {\left (x^{4} - 8 \, x^{3} + 16 \, x^{2}\right )} \log \left (2\right )^{4} + 2 \, {\left (x^{7} - 4 \, x^{6}\right )} \log \left (2\right )^{2}}{\log \left (2\right )^{4}}\right )}}{\log \left (2\right )^{4}} \,d x } \]
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Timed out. \[ \int \frac {e^{-\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} \left (-30 x^9+\left (144 x^5-42 x^6\right ) \log ^2(2)+2 e^{x^2+\frac {x^{10}+\left (-8 x^6+2 x^7\right ) \log ^2(2)+\left (16 x^2-8 x^3+x^4\right ) \log ^4(2)}{\log ^4(2)}} x \log ^4(2)+\left (-96 x+72 x^2-12 x^3\right ) \log ^4(2)\right )}{\log ^4(2)} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {{\ln \left (2\right )}^4\,\left (x^4-8\,x^3+16\,x^2\right )+x^{10}-{\ln \left (2\right )}^2\,\left (8\,x^6-2\,x^7\right )}{{\ln \left (2\right )}^4}}\,\left ({\ln \left (2\right )}^4\,\left (12\,x^3-72\,x^2+96\,x\right )+30\,x^9-{\ln \left (2\right )}^2\,\left (144\,x^5-42\,x^6\right )-2\,x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {{\ln \left (2\right )}^4\,\left (x^4-8\,x^3+16\,x^2\right )+x^{10}-{\ln \left (2\right )}^2\,\left (8\,x^6-2\,x^7\right )}{{\ln \left (2\right )}^4}}\,{\ln \left (2\right )}^4\right )}{{\ln \left (2\right )}^4} \,d x \]
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