Integrand size = 150, antiderivative size = 30 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{\left (-x+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} x+x^2\right )^2} \]
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Time = 4.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6820, 6838} \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{\left (-x-e^{\frac {1}{4} \left (e^{e^x+3}+5\right )}+1\right )^2 x^2} \]
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Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int e^{x^2 \left (-1+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}+x\right )^2} \left (1-e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}-x\right ) x \left (4-4 e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}-8 x-e^{\frac {17}{4}+\frac {e^{3+e^x}}{4}+e^x+x} x\right ) \, dx \\ & = 2 e^{\left (1-e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}-x\right )^2 x^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{x^2 \left (-1+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )}+x\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).
Time = 1.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70
method | result | size |
risch | \(2 \,{\mathrm e}^{x^{2} \left (2 x \,{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+x^{2}-2 \,{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+{\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{2}+\frac {5}{2}}-2 x +1\right )}\) | \(51\) |
parallelrisch | \(2 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{2}+\frac {5}{2}}+\left (2 x^{3}-2 x^{2}\right ) {\mathrm e}^{\frac {{\mathrm e}^{3+{\mathrm e}^{x}}}{4}+\frac {5}{4}}+x^{4}-2 x^{3}+x^{2}}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 \, e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, {\left (e^{\left (x + e^{x} + 3\right )} + 5 \, e^{x}\right )} e^{\left (-x\right )}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, {\left (e^{\left (x + e^{x} + 3\right )} + 5 \, e^{x}\right )} e^{\left (-x\right )}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
Time = 3.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2 e^{x^{4} - 2 x^{3} + x^{2} e^{\frac {e^{e^{x} + 3}}{2} + \frac {5}{2}} + x^{2} + \left (2 x^{3} - 2 x^{2}\right ) e^{\frac {e^{e^{x} + 3}}{4} + \frac {5}{4}}} \]
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\[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=\int { {\left (8 \, x^{3} - 12 \, x^{2} + {\left (x^{2} e^{\left (x + e^{x} + 3\right )} + 4 \, x\right )} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + {\left (12 \, x^{2} + {\left (x^{3} - x^{2}\right )} e^{\left (x + e^{x} + 3\right )} - 8 \, x\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )} + 4 \, x\right )} e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )}\right )} \,d x } \]
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\[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=\int { {\left (8 \, x^{3} - 12 \, x^{2} + {\left (x^{2} e^{\left (x + e^{x} + 3\right )} + 4 \, x\right )} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + {\left (12 \, x^{2} + {\left (x^{3} - x^{2}\right )} e^{\left (x + e^{x} + 3\right )} - 8 \, x\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )} + 4 \, x\right )} e^{\left (x^{4} - 2 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, e^{\left (e^{x} + 3\right )} + \frac {5}{2}\right )} + x^{2} + 2 \, {\left (x^{3} - x^{2}\right )} e^{\left (\frac {1}{4} \, e^{\left (e^{x} + 3\right )} + \frac {5}{4}\right )}\right )} \,d x } \]
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Time = 13.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int e^{x^2+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} x^2-2 x^3+x^4+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-2 x^2+2 x^3\right )} \left (4 x-12 x^2+8 x^3+e^{\frac {1}{2} \left (5+e^{3+e^x}\right )} \left (4 x+e^{3+e^x+x} x^2\right )+e^{\frac {1}{4} \left (5+e^{3+e^x}\right )} \left (-8 x+12 x^2+e^{3+e^x+x} \left (-x^2+x^3\right )\right )\right ) \, dx=2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{5/2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{2}}}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{5/4}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{4}}}\,{\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^{5/4}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{4}}}\,{\mathrm {e}}^{-2\,x^3} \]
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