Integrand size = 31, antiderivative size = 28 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=\frac {1}{5} \left (3+e^{-x} \left (4+x \left (-1+\frac {x}{2}-x \log (3)\right )\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64, number of steps used = 16, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {12, 2227, 2225, 2207} \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=\frac {1}{10} e^{-x} x^2-\frac {1}{5} e^{-x} x^2 \log (3)-\frac {e^{-x} x}{5}+\frac {4 e^{-x}}{5} \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{10} \int e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx \\ & = \frac {1}{10} \int \left (-10 e^{-x}+4 e^{-x} x-e^{-x} x^2+2 e^{-x} (-2+x) x \log (3)\right ) \, dx \\ & = -\left (\frac {1}{10} \int e^{-x} x^2 \, dx\right )+\frac {2}{5} \int e^{-x} x \, dx+\frac {1}{5} \log (3) \int e^{-x} (-2+x) x \, dx-\int e^{-x} \, dx \\ & = e^{-x}-\frac {2 e^{-x} x}{5}+\frac {1}{10} e^{-x} x^2-\frac {1}{5} \int e^{-x} x \, dx+\frac {2}{5} \int e^{-x} \, dx+\frac {1}{5} \log (3) \int \left (-2 e^{-x} x+e^{-x} x^2\right ) \, dx \\ & = \frac {3 e^{-x}}{5}-\frac {e^{-x} x}{5}+\frac {1}{10} e^{-x} x^2-\frac {1}{5} \int e^{-x} \, dx+\frac {1}{5} \log (3) \int e^{-x} x^2 \, dx-\frac {1}{5} (2 \log (3)) \int e^{-x} x \, dx \\ & = \frac {4 e^{-x}}{5}-\frac {e^{-x} x}{5}+\frac {1}{10} e^{-x} x^2+\frac {2}{5} e^{-x} x \log (3)-\frac {1}{5} e^{-x} x^2 \log (3)-\frac {1}{5} (2 \log (3)) \int e^{-x} \, dx+\frac {1}{5} (2 \log (3)) \int e^{-x} x \, dx \\ & = \frac {4 e^{-x}}{5}-\frac {e^{-x} x}{5}+\frac {1}{10} e^{-x} x^2+\frac {2}{5} e^{-x} \log (3)-\frac {1}{5} e^{-x} x^2 \log (3)+\frac {1}{5} (2 \log (3)) \int e^{-x} \, dx \\ & = \frac {4 e^{-x}}{5}-\frac {e^{-x} x}{5}+\frac {1}{10} e^{-x} x^2-\frac {1}{5} e^{-x} x^2 \log (3) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=\frac {1}{10} e^{-x} \left (8-2 x-x^2 (-1+\log (9))\right ) \]
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Time = 0.10 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.75
method | result | size |
norman | \(\left (\frac {4}{5}+\left (-\frac {\ln \left (3\right )}{5}+\frac {1}{10}\right ) x^{2}-\frac {x}{5}\right ) {\mathrm e}^{-x}\) | \(21\) |
risch | \(\frac {\left (-2 x^{2} \ln \left (3\right )+x^{2}-2 x +8\right ) {\mathrm e}^{-x}}{10}\) | \(22\) |
gosper | \(-\frac {\left (2 x^{2} \ln \left (3\right )-x^{2}+2 x -8\right ) {\mathrm e}^{-x}}{10}\) | \(24\) |
parallelrisch | \(-\frac {\left (2 x^{2} \ln \left (3\right )-x^{2}+2 x -8\right ) {\mathrm e}^{-x}}{10}\) | \(24\) |
meijerg | \(-1+{\mathrm e}^{-x}+\left (\frac {\ln \left (3\right )}{5}-\frac {1}{10}\right ) \left (2-\frac {\left (3 x^{2}+6 x +6\right ) {\mathrm e}^{-x}}{3}\right )+\left (-\frac {2 \ln \left (3\right )}{5}+\frac {2}{5}\right ) \left (1-\frac {\left (2+2 x \right ) {\mathrm e}^{-x}}{2}\right )\) | \(52\) |
default | \(\frac {4 \,{\mathrm e}^{-x}}{5}-\frac {x \,{\mathrm e}^{-x}}{5}+\frac {x^{2} {\mathrm e}^{-x}}{10}-\frac {2 \ln \left (3\right ) \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )}{5}+\frac {\ln \left (3\right ) \left (-x^{2} {\mathrm e}^{-x}-2 x \,{\mathrm e}^{-x}-2 \,{\mathrm e}^{-x}\right )}{5}\) | \(69\) |
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Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=-\frac {1}{10} \, {\left (2 \, x^{2} \log \left (3\right ) - x^{2} + 2 \, x - 8\right )} e^{\left (-x\right )} \]
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Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=\frac {\left (- 2 x^{2} \log {\left (3 \right )} + x^{2} - 2 x + 8\right ) e^{- x}}{10} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=-\frac {1}{5} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} \log \left (3\right ) + \frac {2}{5} \, {\left (x + 1\right )} e^{\left (-x\right )} \log \left (3\right ) + \frac {1}{10} \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} - \frac {2}{5} \, {\left (x + 1\right )} e^{\left (-x\right )} + e^{\left (-x\right )} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=-\frac {1}{10} \, {\left (2 \, x^{2} \log \left (3\right ) - x^{2} + 2 \, x - 8\right )} e^{\left (-x\right )} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {1}{10} e^{-x} \left (-10+4 x-x^2+\left (-4 x+2 x^2\right ) \log (3)\right ) \, dx=-\frac {{\mathrm {e}}^{-x}\,\left (2\,x+x^2\,\ln \left (9\right )-x^2-8\right )}{10} \]
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