Integrand size = 167, antiderivative size = 24 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{\left (4-5 e^x+e^{\log ^4(2)}-x\right )^2 x^2} \]
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Time = 7.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 6820, 12, 6838} \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{x^2 \left (-x-5 e^x+4+e^{\log ^4(2)}\right )^2} \]
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Rule 6
Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \exp \left (16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )\right ) \left (\left (32+2 e^{2 \log ^4(2)}\right ) x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx \\ & = \int 2 \exp \left (\left (-5 e^x+4 \left (1+\frac {1}{4} e^{\log ^4(2)}\right )-x\right )^2 x^2\right ) x \left (16 \left (1+\frac {1}{16} e^{2 \log ^4(2)}\right )-12 x+2 x^2+25 e^{2 x} (1+x)+5 e^x \left (-8-x+x^2\right )+e^{\log ^4(2)} \left (8-3 x-5 e^x (2+x)\right )\right ) \, dx \\ & = 2 \int \exp \left (\left (-5 e^x+4 \left (1+\frac {1}{4} e^{\log ^4(2)}\right )-x\right )^2 x^2\right ) x \left (16 \left (1+\frac {1}{16} e^{2 \log ^4(2)}\right )-12 x+2 x^2+25 e^{2 x} (1+x)+5 e^x \left (-8-x+x^2\right )+e^{\log ^4(2)} \left (8-3 x-5 e^x (2+x)\right )\right ) \, dx \\ & = e^{\left (4-5 e^x+e^{\log ^4(2)}-x\right )^2 x^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{\left (4-5 e^x+e^{\log ^4(2)}-x\right )^2 x^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(21)=42\).
Time = 0.86 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.50
method | result | size |
risch | \({\mathrm e}^{x^{2} \left (-10 \,{\mathrm e}^{x +\ln \left (2\right )^{4}}+10 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{\ln \left (2\right )^{4}} x +x^{2}+{\mathrm e}^{2 \ln \left (2\right )^{4}}-40 \,{\mathrm e}^{x}+25 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{\ln \left (2\right )^{4}}-8 x +16\right )}\) | \(60\) |
norman | \({\mathrm e}^{x^{2} {\mathrm e}^{2 \ln \left (2\right )^{4}}+\left (-10 \,{\mathrm e}^{x} x^{2}-2 x^{3}+8 x^{2}\right ) {\mathrm e}^{\ln \left (2\right )^{4}}+25 \,{\mathrm e}^{2 x} x^{2}+\left (10 x^{3}-40 x^{2}\right ) {\mathrm e}^{x}+x^{4}-8 x^{3}+16 x^{2}}\) | \(74\) |
parallelrisch | \({\mathrm e}^{x^{2} {\mathrm e}^{2 \ln \left (2\right )^{4}}+\left (-10 \,{\mathrm e}^{x} x^{2}-2 x^{3}+8 x^{2}\right ) {\mathrm e}^{\ln \left (2\right )^{4}}+25 \,{\mathrm e}^{2 x} x^{2}+\left (10 x^{3}-40 x^{2}\right ) {\mathrm e}^{x}+x^{4}-8 x^{3}+16 x^{2}}\) | \(74\) |
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{\left (x^{4} - 8 \, x^{3} + x^{2} e^{\left (2 \, \log \left (2\right )^{4}\right )} + 25 \, x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} - 2 \, {\left (x^{3} + 5 \, x^{2} e^{x} - 4 \, x^{2}\right )} e^{\left (\log \left (2\right )^{4}\right )} + 10 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{x}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.12 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{x^{4} - 8 x^{3} + 25 x^{2} e^{2 x} + x^{2} e^{2 \log {\left (2 \right )}^{4}} + 16 x^{2} + \left (10 x^{3} - 40 x^{2}\right ) e^{x} + \left (- 2 x^{3} - 10 x^{2} e^{x} + 8 x^{2}\right ) e^{\log {\left (2 \right )}^{4}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
Time = 0.48 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{\left (x^{4} - 2 \, x^{3} e^{\left (\log \left (2\right )^{4}\right )} + 10 \, x^{3} e^{x} - 8 \, x^{3} + x^{2} e^{\left (2 \, \log \left (2\right )^{4}\right )} - 10 \, x^{2} e^{\left (\log \left (2\right )^{4} + x\right )} + 8 \, x^{2} e^{\left (\log \left (2\right )^{4}\right )} + 25 \, x^{2} e^{\left (2 \, x\right )} - 40 \, x^{2} e^{x} + 16 \, x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (21) = 42\).
Time = 0.63 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx=e^{\left (x^{4} - 2 \, x^{3} e^{\left (\log \left (2\right )^{4}\right )} + 10 \, x^{3} e^{x} - 8 \, x^{3} + x^{2} e^{\left (2 \, \log \left (2\right )^{4}\right )} - 10 \, x^{2} e^{\left (\log \left (2\right )^{4} + x\right )} + 8 \, x^{2} e^{\left (\log \left (2\right )^{4}\right )} + 25 \, x^{2} e^{\left (2 \, x\right )} - 40 \, x^{2} e^{x} + 16 \, x^{2}\right )} \]
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Time = 14.40 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75 \[ \int e^{16 x^2+25 e^{2 x} x^2+e^{2 \log ^4(2)} x^2-8 x^3+x^4+e^{\log ^4(2)} \left (8 x^2-10 e^x x^2-2 x^3\right )+e^x \left (-40 x^2+10 x^3\right )} \left (32 x+2 e^{2 \log ^4(2)} x-24 x^2+4 x^3+e^{2 x} \left (50 x+50 x^2\right )+e^x \left (-80 x-10 x^2+10 x^3\right )+e^{\log ^4(2)} \left (16 x-6 x^2+e^x \left (-20 x-10 x^2\right )\right )\right ) \, dx={\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,{\ln \left (2\right )}^4}}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{10\,x^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-40\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-8\,x^3}\,{\mathrm {e}}^{16\,x^2}\,{\mathrm {e}}^{-10\,x^2\,{\mathrm {e}}^{{\ln \left (2\right )}^4}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-2\,x^3\,{\mathrm {e}}^{{\ln \left (2\right )}^4}}\,{\mathrm {e}}^{8\,x^2\,{\mathrm {e}}^{{\ln \left (2\right )}^4}}\,{\mathrm {e}}^{25\,x^2\,{\mathrm {e}}^{2\,x}} \]
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