Integrand size = 39, antiderivative size = 16 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=1+\left (e^{-1+x}+x-x^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(16)=32\).
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.31, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2225, 2227, 2207} \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=x^4-2 x^3-2 e^{x-1} x^2+x^2+2 e^{x-1} x+e^{2 x-2} \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = x^2-2 x^3+x^4+2 \int e^{-2+2 x} \, dx+\int e^{-1+x} \left (2-2 x-2 x^2\right ) \, dx \\ & = e^{-2+2 x}+x^2-2 x^3+x^4+\int \left (2 e^{-1+x}-2 e^{-1+x} x-2 e^{-1+x} x^2\right ) \, dx \\ & = e^{-2+2 x}+x^2-2 x^3+x^4+2 \int e^{-1+x} \, dx-2 \int e^{-1+x} x \, dx-2 \int e^{-1+x} x^2 \, dx \\ & = 2 e^{-1+x}+e^{-2+2 x}-2 e^{-1+x} x+x^2-2 e^{-1+x} x^2-2 x^3+x^4+2 \int e^{-1+x} \, dx+4 \int e^{-1+x} x \, dx \\ & = 4 e^{-1+x}+e^{-2+2 x}+2 e^{-1+x} x+x^2-2 e^{-1+x} x^2-2 x^3+x^4-4 \int e^{-1+x} \, dx \\ & = e^{-2+2 x}+2 e^{-1+x} x+x^2-2 e^{-1+x} x^2-2 x^3+x^4 \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=\frac {\left (e^x-e (-1+x) x\right )^2}{e^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(32\) vs. \(2(15)=30\).
Time = 0.14 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06
method | result | size |
risch | \({\mathrm e}^{-2+2 x}+\left (-2 x^{2}+2 x \right ) {\mathrm e}^{-1+x}+x^{4}-2 x^{3}+x^{2}\) | \(33\) |
norman | \(x^{4}-2 x^{2} {\mathrm e}^{-1+x}-2 x^{3}+2 x \,{\mathrm e}^{-1+x}+x^{2}+{\mathrm e}^{-2+2 x}\) | \(35\) |
parallelrisch | \(x^{4}-2 x^{2} {\mathrm e}^{-1+x}-2 x^{3}+2 x \,{\mathrm e}^{-1+x}+x^{2}+{\mathrm e}^{-2+2 x}\) | \(35\) |
default | \(-2 \,{\mathrm e}^{-1+x} \left (-1+x \right )-2 \,{\mathrm e}^{-1+x} \left (-1+x \right )^{2}+x^{2}-2 x^{3}+x^{4}+{\mathrm e}^{-2+2 x}\) | \(39\) |
parts | \(-2 \,{\mathrm e}^{-1+x} \left (-1+x \right )-2 \,{\mathrm e}^{-1+x} \left (-1+x \right )^{2}+x^{2}-2 x^{3}+x^{4}+{\mathrm e}^{-2+2 x}\) | \(39\) |
derivativedivides | \(-2+2 x -2 \,{\mathrm e}^{-1+x} \left (-1+x \right )-2 \,{\mathrm e}^{-1+x} \left (-1+x \right )^{2}+\left (-1+x \right )^{2}-2 x^{3}+x^{4}+{\mathrm e}^{-2+2 x}\) | \(45\) |
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Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=x^{4} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} - x\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (12) = 24\).
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=x^{4} - 2 x^{3} + x^{2} + \left (- 2 x^{2} + 2 x\right ) e^{x - 1} + e^{2 x - 2} \]
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Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=x^{4} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} - x\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 2\right )} \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.94 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx=x^{4} - 2 \, x^{3} + x^{2} - 2 \, {\left (x^{2} - x\right )} e^{\left (x - 1\right )} + e^{\left (2 \, x - 2\right )} \]
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Time = 0.15 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \left (2 e^{-2+2 x}+2 x-6 x^2+4 x^3+e^{-1+x} \left (2-2 x-2 x^2\right )\right ) \, dx={\left (x+{\mathrm {e}}^{x-1}-x^2\right )}^2 \]
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