Integrand size = 29, antiderivative size = 26 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=x+e^{\frac {2 e^2 (4-x) x}{3 (-4+x)}} x^2 \]
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Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=e^{-\frac {2 e^2 x}{3}} x^2+x \]
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx \\ & = x+\frac {1}{3} \int e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right ) \, dx \\ & = x+\frac {1}{3} \int e^{-\frac {2 e^2 x}{3}} x \left (6-2 e^2 x\right ) \, dx \\ & = x+\frac {1}{3} \int \left (6 e^{-\frac {2 e^2 x}{3}} x-2 e^{2-\frac {2 e^2 x}{3}} x^2\right ) \, dx \\ & = x-\frac {2}{3} \int e^{2-\frac {2 e^2 x}{3}} x^2 \, dx+2 \int e^{-\frac {2 e^2 x}{3}} x \, dx \\ & = x-3 e^{-2-\frac {2 e^2 x}{3}} x+e^{-\frac {2 e^2 x}{3}} x^2-\frac {2 \int e^{2-\frac {2 e^2 x}{3}} x \, dx}{e^2}+\frac {3 \int e^{-\frac {2 e^2 x}{3}} \, dx}{e^2} \\ & = -\frac {9}{2} e^{-4-\frac {2 e^2 x}{3}}+x+e^{-\frac {2 e^2 x}{3}} x^2-\frac {3 \int e^{2-\frac {2 e^2 x}{3}} \, dx}{e^4} \\ & = x+e^{-\frac {2 e^2 x}{3}} x^2 \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=x+e^{-\frac {2 e^2 x}{3}} x^2 \]
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Time = 0.35 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.50
method | result | size |
risch | \(x +x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}\) | \(13\) |
norman | \(x +x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}\) | \(15\) |
parallelrisch | \(x +x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}\) | \(15\) |
default | \(x -18 \,{\mathrm e}^{-2} \left (-{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )-{\mathrm e}^{-2} \left (\frac {{\mathrm e}^{4} x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{18}+\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )\right )\) | \(87\) |
parts | \(x -18 \,{\mathrm e}^{-2} \left (-{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )-{\mathrm e}^{-2} \left (\frac {{\mathrm e}^{4} x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{18}+\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )\right )\) | \(87\) |
derivativedivides | \(-{\mathrm e}^{-2} \left (-{\mathrm e}^{2} x -18 \,{\mathrm e}^{-2} \left (-\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}-\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )-18 \,{\mathrm e}^{-2} \left (\frac {{\mathrm e}^{4} x^{2} {\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{18}+\frac {{\mathrm e}^{2} x \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{6}+\frac {{\mathrm e}^{-\frac {2 \,{\mathrm e}^{2} x}{3}}}{4}\right )\right )\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.46 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=x^{2} e^{\left (-\frac {2}{3} \, x e^{2}\right )} + x \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=x^{2} e^{- \frac {2 x e^{2}}{3}} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (12) = 24\).
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=\frac {1}{2} \, {\left (2 \, x^{2} e^{4} + 6 \, x e^{2} + 9\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} - \frac {3}{2} \, {\left (2 \, x e^{2} + 3\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} + x \]
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Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=\frac {1}{2} \, {\left (2 \, x^{2} e^{4} + 6 \, x e^{2} + 9\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} - \frac {3}{2} \, {\left (2 \, x e^{2} + 3\right )} e^{\left (-\frac {2}{3} \, x e^{2} - 4\right )} + x \]
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Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.46 \[ \int \frac {1}{3} \left (3+e^{-\frac {2 e^2 x}{3}} \left (6 x-2 e^2 x^2\right )\right ) \, dx=x+x^2\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^2}{3}} \]
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