Integrand size = 178, antiderivative size = 26 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\frac {2 \left (e^{2 x}+\left (e^x+x^2\right )^2\right )^2}{\sqrt {e}}} \]
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Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(26)=52\).
Time = 3.71 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6820, 12, 6838} \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=\exp \left (\frac {1}{2}-\frac {-4 x^8-16 e^x x^6-32 e^{2 x} x^4-32 e^{3 x} x^2-16 e^{4 x}+\sqrt {e}}{2 \sqrt {e}}\right ) \]
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Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int 8 \exp \left (\frac {-\sqrt {e}+16 e^{4 x}+32 e^{3 x} x^2+32 e^{2 x} x^4+16 e^x x^6+4 x^8}{2 \sqrt {e}}\right ) \left (4 e^{4 x}+2 x^7+4 e^{2 x} x^3 (2+x)+e^x x^5 (6+x)+2 e^{3 x} x (2+3 x)\right ) \, dx \\ & = 8 \int \exp \left (\frac {-\sqrt {e}+16 e^{4 x}+32 e^{3 x} x^2+32 e^{2 x} x^4+16 e^x x^6+4 x^8}{2 \sqrt {e}}\right ) \left (4 e^{4 x}+2 x^7+4 e^{2 x} x^3 (2+x)+e^x x^5 (6+x)+2 e^{3 x} x (2+3 x)\right ) \, dx \\ & = \exp \left (\frac {1}{2}-\frac {\sqrt {e}-16 e^{4 x}-32 e^{3 x} x^2-32 e^{2 x} x^4-16 e^x x^6-4 x^8}{2 \sqrt {e}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(26)=52\).
Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{8 e^{-\frac {1}{2}+4 x}+16 e^{-\frac {1}{2}+3 x} x^2+16 e^{-\frac {1}{2}+2 x} x^4+8 e^{-\frac {1}{2}+x} x^6+\frac {2 x^8}{\sqrt {e}}} \]
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Time = 14.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58
method | result | size |
risch | \({\mathrm e}^{2 \left (x^{8}+4 x^{6} {\mathrm e}^{x}+8 \,{\mathrm e}^{2 x} x^{4}+8 x^{2} {\mathrm e}^{3 x}+4 \,{\mathrm e}^{4 x}\right ) {\mathrm e}^{-\frac {1}{2}}}\) | \(41\) |
parallelrisch | \({\mathrm e}^{\left (4 \,{\mathrm e}^{4 x}+\left (4 \,{\mathrm e}^{2 x}+8 \,{\mathrm e}^{x} x^{2}+4 x^{4}\right ) {\mathrm e}^{2 x}+8 x^{2} {\mathrm e}^{3 x}+12 \,{\mathrm e}^{2 x} x^{4}+8 x^{6} {\mathrm e}^{x}+2 x^{8}\right ) {\mathrm e}^{-\frac {1}{2}}}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (\frac {1}{2} \, {\left (4 \, x^{8} + 16 \, x^{6} e^{x} + 32 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{2} e^{\left (3 \, x\right )} - e^{\frac {1}{2}} + 16 \, e^{\left (4 \, x\right )}\right )} e^{\left (-\frac {1}{2}\right )} + \frac {1}{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\frac {2 x^{8} + 8 x^{6} e^{x} + 12 x^{4} e^{2 x} + 8 x^{2} e^{3 x} + \left (4 x^{4} + 8 x^{2} e^{x} + 4 e^{2 x}\right ) e^{2 x} + 4 e^{4 x}}{e^{\frac {1}{2}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.41 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (2 \, x^{8} e^{\left (-\frac {1}{2}\right )} + 8 \, x^{6} e^{\left (x - \frac {1}{2}\right )} + 16 \, x^{4} e^{\left (2 \, x - \frac {1}{2}\right )} + 16 \, x^{2} e^{\left (3 \, x - \frac {1}{2}\right )} + 8 \, e^{\left (4 \, x - \frac {1}{2}\right )}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.35 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx=e^{\left (2 \, {\left (x^{8} e^{\left (4 \, x - 2\right )} + 4 \, x^{6} e^{\left (5 \, x - 2\right )} + 8 \, x^{4} e^{\left (6 \, x - 2\right )} + 8 \, x^{2} e^{\left (7 \, x - 2\right )} + 4 \, e^{\left (8 \, x - 2\right )}\right )} e^{\left (-4 \, x + \frac {3}{2}\right )}\right )} \]
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Time = 13.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int e^{-\frac {1}{2}+\frac {4 e^{4 x}+8 e^{3 x} x^2+12 e^{2 x} x^4+8 e^x x^6+2 x^8+e^{2 x} \left (4 e^{2 x}+8 e^x x^2+4 x^4\right )}{\sqrt {e}}} \left (16 e^{4 x}+16 x^7+e^{3 x} \left (16 x+24 x^2\right )+e^{2 x} \left (48 x^3+24 x^4\right )+e^x \left (48 x^5+8 x^6\right )+e^{2 x} \left (16 e^{2 x}+16 x^3+8 x^4+e^x \left (16 x+24 x^2\right )\right )\right ) \, dx={\mathrm {e}}^{2\,x^8\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{16\,x^2\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{16\,x^4\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{8\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{-\frac {1}{2}}}\,{\mathrm {e}}^{8\,x^6\,{\mathrm {e}}^{-\frac {1}{2}}\,{\mathrm {e}}^x} \]
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