Integrand size = 107, antiderivative size = 30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=2+3 \left (e^{e^{e^{2-x}}}+\left (3+\frac {2}{-3+x+x^2}\right )^2\right ) \]
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Time = 0.31 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6820, 2320, 2225, 1674, 643} \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=-\frac {36}{-x^2-x+3}+\frac {12}{\left (-x^2-x+3\right )^2}+3 e^{e^{e^{2-x}}} \]
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Rule 643
Rule 1674
Rule 2225
Rule 2320
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \left (-3 e^{2+e^{e^{2-x}}+e^{2-x}-x}-\frac {12 \left (-7-11 x+9 x^2+6 x^3\right )}{\left (-3+x+x^2\right )^3}\right ) \, dx \\ & = -\left (3 \int e^{2+e^{e^{2-x}}+e^{2-x}-x} \, dx\right )-12 \int \frac {-7-11 x+9 x^2+6 x^3}{\left (-3+x+x^2\right )^3} \, dx \\ & = \frac {12}{\left (3-x-x^2\right )^2}+\frac {6}{13} \int \frac {-78-156 x}{\left (-3+x+x^2\right )^2} \, dx+3 \text {Subst}\left (\int e^{2+e^{e^2 x}+e^2 x} \, dx,x,e^{-x}\right ) \\ & = \frac {12}{\left (3-x-x^2\right )^2}-\frac {36}{3-x-x^2}+\frac {3 \text {Subst}\left (\int e^{2+x} \, dx,x,e^{e^{2-x}}\right )}{e^2} \\ & = 3 e^{e^{e^{2-x}}}+\frac {12}{\left (3-x-x^2\right )^2}-\frac {36}{3-x-x^2} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=3 e^{e^{e^{2-x}}}+\frac {12}{\left (-3+x+x^2\right )^2}+\frac {36}{-3+x+x^2} \]
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Time = 34.52 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
parts | \(-\frac {12 \left (-3 x^{2}-3 x +8\right )}{\left (x^{2}+x -3\right )^{2}}+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}\) | \(33\) |
risch | \(\frac {36 x^{2}+36 x -96}{x^{4}+2 x^{3}-5 x^{2}-6 x +9}+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2-x}}}\) | \(43\) |
parallelrisch | \(\frac {-96+3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x^{4}+6 x^{3} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}-15 x^{2} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}+36 x^{2}-18 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}} x +36 x +27 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2} {\mathrm e}^{-x}}}}{x^{4}+2 x^{3}-5 x^{2}-6 x +9}\) | \(97\) |
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Timed out. \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=- \frac {- 36 x^{2} - 36 x + 96}{x^{4} + 2 x^{3} - 5 x^{2} - 6 x + 9} + 3 e^{e^{e^{2} e^{- x}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (28) = 56\).
Time = 0.52 (sec) , antiderivative size = 159, normalized size of antiderivative = 5.30 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=-\frac {36 \, {\left (18 \, x^{3} - 142 \, x^{2} - 84 \, x + 207\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + \frac {42 \, {\left (12 \, x^{3} + 18 \, x^{2} - 56 \, x - 31\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + \frac {54 \, {\left (10 \, x^{3} + 15 \, x^{2} + 66 \, x - 54\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} - \frac {66 \, {\left (6 \, x^{3} + 9 \, x^{2} - 28 \, x + 69\right )}}{169 \, {\left (x^{4} + 2 \, x^{3} - 5 \, x^{2} - 6 \, x + 9\right )}} + 3 \, e^{\left (e^{\left (e^{\left (-x + 2\right )}\right )}\right )} \]
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\[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=\int { -\frac {3 \, {\left (4 \, {\left (6 \, x^{3} + 9 \, x^{2} - 11 \, x - 7\right )} e^{x} + {\left (x^{6} + 3 \, x^{5} - 6 \, x^{4} - 17 \, x^{3} + 18 \, x^{2} + 27 \, x - 27\right )} e^{\left (e^{\left (-x + 2\right )} + e^{\left (e^{\left (-x + 2\right )}\right )} + 2\right )}\right )} e^{\left (-x\right )}}{x^{6} + 3 \, x^{5} - 6 \, x^{4} - 17 \, x^{3} + 18 \, x^{2} + 27 \, x - 27} \,d x } \]
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Time = 10.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-x} \left (e^x \left (84+132 x-108 x^2-72 x^3\right )+e^{2+e^{e^{2-x}}+e^{2-x}} \left (81-81 x-54 x^2+51 x^3+18 x^4-9 x^5-3 x^6\right )\right )}{-27+27 x+18 x^2-17 x^3-6 x^4+3 x^5+x^6} \, dx=3\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{-x}\,{\mathrm {e}}^2}}+\frac {36\,x^2+36\,x-96}{{\left (x^2+x-3\right )}^2} \]
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