Integrand size = 80, antiderivative size = 26 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=1-e^{\left (-3+\frac {3}{-1+x}\right )^2 x^2}-2 x+\log (2) \]
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Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(26)=52\).
Time = 0.48 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6820, 6874, 37, 45, 6838} \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=\frac {3 x^2}{(1-x)^2}-e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-2 x+\frac {6}{1-x}-\frac {3}{(1-x)^2} \]
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Rule 37
Rule 45
Rule 6820
Rule 6838
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2+6 x-6 x^2+2 x^3+18 e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} x \left (-4+6 x-4 x^2+x^3\right )}{(1-x)^3} \, dx \\ & = \int \left (\frac {2}{(-1+x)^3}-\frac {6 x}{(-1+x)^3}+\frac {6 x^2}{(-1+x)^3}-\frac {2 x^3}{(-1+x)^3}-\frac {18 e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} (-2+x) x \left (2-2 x+x^2\right )}{(-1+x)^3}\right ) \, dx \\ & = -\frac {1}{(1-x)^2}-2 \int \frac {x^3}{(-1+x)^3} \, dx-6 \int \frac {x}{(-1+x)^3} \, dx+6 \int \frac {x^2}{(-1+x)^3} \, dx-18 \int \frac {e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} (-2+x) x \left (2-2 x+x^2\right )}{(-1+x)^3} \, dx \\ & = -e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-\frac {1}{(1-x)^2}+\frac {3 x^2}{(1-x)^2}-2 \int \left (1+\frac {1}{(-1+x)^3}+\frac {3}{(-1+x)^2}+\frac {3}{-1+x}\right ) \, dx+6 \int \left (\frac {1}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx \\ & = -e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-\frac {3}{(1-x)^2}+\frac {6}{1-x}-2 x+\frac {3 x^2}{(1-x)^2} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=-e^{-9+\frac {9}{(-1+x)^2}-18 x+9 x^2}-2 x \]
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Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-2 x -{\mathrm e}^{\frac {9 x^{2} \left (-2+x \right )^{2}}{\left (-1+x \right )^{2}}}\) | \(23\) |
parallelrisch | \(-2 x -{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-12\) | \(36\) |
parts | \(-2 x +\frac {2 x \,{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-x^{2} {\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}}{\left (-1+x \right )^{2}}\) | \(106\) |
norman | \(\frac {6 x -2 x^{3}+2 x \,{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-x^{2} {\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-4}{\left (-1+x \right )^{2}}\) | \(111\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=-2 \, x - e^{\left (\frac {9 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}}{x^{2} - 2 \, x + 1}\right )} \]
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Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=- 2 x - e^{\frac {9 x^{4} - 36 x^{3} + 36 x^{2}}{x^{2} - 2 x + 1}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.50 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=-2 \, x + \frac {6 \, x - 5}{x^{2} - 2 \, x + 1} - \frac {3 \, {\left (4 \, x - 3\right )}}{x^{2} - 2 \, x + 1} + \frac {3 \, {\left (2 \, x - 1\right )}}{x^{2} - 2 \, x + 1} - \frac {1}{x^{2} - 2 \, x + 1} - e^{\left (9 \, x^{2} - 18 \, x + \frac {9}{x^{2} - 2 \, x + 1} - 9\right )} \]
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Time = 0.35 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=-2 \, x - e^{\left (\frac {9 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}}{x^{2} - 2 \, x + 1}\right )} \]
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Time = 11.79 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} \left (72 x-108 x^2+72 x^3-18 x^4\right )}{-1+3 x-3 x^2+x^3} \, dx=-2\,x-{\mathrm {e}}^{\frac {9\,x^4}{x^2-2\,x+1}}\,{\mathrm {e}}^{\frac {36\,x^2}{x^2-2\,x+1}}\,{\mathrm {e}}^{-\frac {36\,x^3}{x^2-2\,x+1}} \]
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