Integrand size = 201, antiderivative size = 31 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{x^2 \left (2 \left (3+e^{\frac {x^2}{e}}\right )-x+x \log ^2(25)\right )^2} \]
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Time = 3.97 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6820, 12, 6838} \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=\exp \left (x^4 \left (1-\log ^2(25)\right )^2+4 \left (e^{\frac {x^2}{e}}+3\right )^2 x^2-4 \left (e^{\frac {x^2}{e}}+3\right ) x^3 \left (1-\log ^2(25)\right )\right ) \]
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Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int 4 \exp \left (-1+4 \left (3+e^{\frac {x^2}{e}}\right )^2 x^2+4 \left (3+e^{\frac {x^2}{e}}\right ) x^3 \left (-1+\log ^2(25)\right )+x^4 \left (-1+\log ^2(25)\right )^2\right ) x \left (6+2 e^{\frac {x^2}{e}}+x \left (-1+\log ^2(25)\right )\right ) \left (e^{1+\frac {x^2}{e}}+2 e^{\frac {x^2}{e}} x^2+e \left (3+x \left (-1+\log ^2(25)\right )\right )\right ) \, dx \\ & = 4 \int \exp \left (-1+4 \left (3+e^{\frac {x^2}{e}}\right )^2 x^2+4 \left (3+e^{\frac {x^2}{e}}\right ) x^3 \left (-1+\log ^2(25)\right )+x^4 \left (-1+\log ^2(25)\right )^2\right ) x \left (6+2 e^{\frac {x^2}{e}}+x \left (-1+\log ^2(25)\right )\right ) \left (e^{1+\frac {x^2}{e}}+2 e^{\frac {x^2}{e}} x^2+e \left (3+x \left (-1+\log ^2(25)\right )\right )\right ) \, dx \\ & = \exp \left (4 \left (3+e^{\frac {x^2}{e}}\right )^2 x^2-4 \left (3+e^{\frac {x^2}{e}}\right ) x^3 \left (1-\log ^2(25)\right )+x^4 \left (1-\log ^2(25)\right )^2\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{x^2 \left (6+2 e^{\frac {x^2}{e}}+x \left (-1+\log ^2(25)\right )\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(30)=60\).
Time = 5.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65
method | result | size |
risch | \({\mathrm e}^{x^{2} \left (16 \ln \left (5\right )^{4} x^{2}+16 \ln \left (5\right )^{2} {\mathrm e}^{x^{2} {\mathrm e}^{-1}} x -8 x^{2} \ln \left (5\right )^{2}+48 x \ln \left (5\right )^{2}-4 \,{\mathrm e}^{x^{2} {\mathrm e}^{-1}} x +x^{2}+4 \,{\mathrm e}^{2 x^{2} {\mathrm e}^{-1}}+24 \,{\mathrm e}^{x^{2} {\mathrm e}^{-1}}-12 x +36\right )}\) | \(82\) |
norman | \({\mathrm e}^{4 x^{2} {\mathrm e}^{2 x^{2} {\mathrm e}^{-1}}+\left (16 x^{3} \ln \left (5\right )^{2}-4 x^{3}+24 x^{2}\right ) {\mathrm e}^{x^{2} {\mathrm e}^{-1}}+16 x^{4} \ln \left (5\right )^{4}+4 \left (-2 x^{4}+12 x^{3}\right ) \ln \left (5\right )^{2}+x^{4}-12 x^{3}+36 x^{2}}\) | \(88\) |
parallelrisch | \({\mathrm e}^{4 x^{2} {\mathrm e}^{2 x^{2} {\mathrm e}^{-1}}+\left (16 x^{3} \ln \left (5\right )^{2}-4 x^{3}+24 x^{2}\right ) {\mathrm e}^{x^{2} {\mathrm e}^{-1}}+16 x^{4} \ln \left (5\right )^{4}+4 \left (-2 x^{4}+12 x^{3}\right ) \ln \left (5\right )^{2}+x^{4}-12 x^{3}+36 x^{2}}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
Time = 0.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.61 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{\left (16 \, x^{4} \log \left (5\right )^{4} + x^{4} - 12 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x^{2} e^{\left (-1\right )}\right )} - 8 \, {\left (x^{4} - 6 \, x^{3}\right )} \log \left (5\right )^{2} + 36 \, x^{2} + 4 \, {\left (4 \, x^{3} \log \left (5\right )^{2} - x^{3} + 6 \, x^{2}\right )} e^{\left (x^{2} e^{\left (-1\right )}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (27) = 54\).
Time = 0.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.68 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{x^{4} + 16 x^{4} \log {\left (5 \right )}^{4} - 12 x^{3} + 4 x^{2} e^{\frac {2 x^{2}}{e}} + 36 x^{2} + \left (- 8 x^{4} + 48 x^{3}\right ) \log {\left (5 \right )}^{2} + \left (- 4 x^{3} + 16 x^{3} \log {\left (5 \right )}^{2} + 24 x^{2}\right ) e^{\frac {x^{2}}{e}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (28) = 56\).
Time = 0.53 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{\left (16 \, x^{4} \log \left (5\right )^{4} - 8 \, x^{4} \log \left (5\right )^{2} + 16 \, x^{3} e^{\left (x^{2} e^{\left (-1\right )}\right )} \log \left (5\right )^{2} + 48 \, x^{3} \log \left (5\right )^{2} + x^{4} - 4 \, x^{3} e^{\left (x^{2} e^{\left (-1\right )}\right )} - 12 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x^{2} e^{\left (-1\right )}\right )} + 24 \, x^{2} e^{\left (x^{2} e^{\left (-1\right )}\right )} + 36 \, x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (28) = 56\).
Time = 0.87 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx=e^{\left (16 \, x^{4} \log \left (5\right )^{4} - 8 \, x^{4} \log \left (5\right )^{2} + 16 \, x^{3} e^{\left (x^{2} e^{\left (-1\right )}\right )} \log \left (5\right )^{2} + 48 \, x^{3} \log \left (5\right )^{2} + x^{4} - 4 \, x^{3} e^{\left (x^{2} e^{\left (-1\right )}\right )} - 12 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x^{2} e^{\left (-1\right )}\right )} + 24 \, x^{2} e^{\left (x^{2} e^{\left (-1\right )}\right )} + 36 \, x^{2}\right )} \]
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Time = 13.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.35 \[ \int e^{-1+36 x^2+4 e^{\frac {2 x^2}{e}} x^2-12 x^3+x^4+\left (12 x^3-2 x^4\right ) \log ^2(25)+x^4 \log ^4(25)+e^{\frac {x^2}{e}} \left (24 x^2-4 x^3+4 x^3 \log ^2(25)\right )} \left (e \left (72 x-36 x^2+4 x^3\right )+e^{\frac {2 x^2}{e}} \left (8 e x+16 x^3\right )+e \left (36 x^2-8 x^3\right ) \log ^2(25)+4 e x^3 \log ^4(25)+e^{\frac {x^2}{e}} \left (48 x^3-8 x^4+e \left (48 x-12 x^2\right )+\left (12 e x^2+8 x^4\right ) \log ^2(25)\right )\right ) \, dx={\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-8\,x^4\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{16\,x^4\,{\ln \left (5\right )}^4}\,{\mathrm {e}}^{48\,x^3\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{16\,x^3\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-1}}\,{\ln \left (5\right )}^2}\,{\mathrm {e}}^{-12\,x^3}\,{\mathrm {e}}^{36\,x^2}\,{\mathrm {e}}^{-4\,x^3\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-1}}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{-1}}}\,{\mathrm {e}}^{24\,x^2\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-1}}} \]
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