Integrand size = 38, antiderivative size = 23 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=e^{-1-3 x} x \left (-\frac {x^2}{4}+\log (2)-\log (16)\right ) \]
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Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 2227, 2207, 2225} \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=-\frac {1}{4} e^{-3 x-1} x^3-\frac {1}{3} e^{-3 x-1} \log (8)+\frac {1}{3} e^{-3 x-1} (1-3 x) \log (8) \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \int e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx \\ & = \frac {1}{4} \int \left (-3 e^{-1-3 x} x^2+3 e^{-1-3 x} x^3-4 e^{-1-3 x} (-1+3 x) \log (2) \left (1-\frac {\log (16)}{\log (2)}\right )\right ) \, dx \\ & = -\left (\frac {3}{4} \int e^{-1-3 x} x^2 \, dx\right )+\frac {3}{4} \int e^{-1-3 x} x^3 \, dx+\log (8) \int e^{-1-3 x} (-1+3 x) \, dx \\ & = \frac {1}{4} e^{-1-3 x} x^2-\frac {1}{4} e^{-1-3 x} x^3+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)-\frac {1}{2} \int e^{-1-3 x} x \, dx+\frac {3}{4} \int e^{-1-3 x} x^2 \, dx+\log (8) \int e^{-1-3 x} \, dx \\ & = \frac {1}{6} e^{-1-3 x} x-\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)-\frac {1}{6} \int e^{-1-3 x} \, dx+\frac {1}{2} \int e^{-1-3 x} x \, dx \\ & = \frac {1}{18} e^{-1-3 x}-\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8)+\frac {1}{6} \int e^{-1-3 x} \, dx \\ & = -\frac {1}{4} e^{-1-3 x} x^3-\frac {1}{3} e^{-1-3 x} \log (8)+\frac {1}{3} e^{-1-3 x} (1-3 x) \log (8) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=\frac {1}{4} e^{-1-3 x} \left (-x^3-4 x \log (8)\right ) \]
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Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(-\frac {{\mathrm e}^{-3 x -1} x \left (x^{2}+12 \ln \left (2\right )\right )}{4}\) | \(18\) |
risch | \(\frac {\left (-x^{3}-12 x \ln \left (2\right )\right ) {\mathrm e}^{-3 x -1}}{4}\) | \(20\) |
norman | \(-\frac {x^{3} {\mathrm e}^{-3 x -1}}{4}-3 x \ln \left (2\right ) {\mathrm e}^{-3 x -1}\) | \(24\) |
parallelrisch | \(-\frac {x^{3} {\mathrm e}^{-3 x -1}}{4}-3 x \ln \left (2\right ) {\mathrm e}^{-3 x -1}\) | \(24\) |
parts | \(-3 x \ln \left (2\right ) {\mathrm e}^{-3 x -1}-\frac {x^{3} {\mathrm e}^{-3 x -1}}{4}+\frac {{\mathrm e}^{-3 x -1} x^{2}}{4}-\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{2}}{36}-\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )}{18}-\frac {{\mathrm e}^{-3 x -1}}{36}\) | \(71\) |
meijerg | \(-\ln \left (2\right ) {\mathrm e}^{-1} \left (1-{\mathrm e}^{-3 x}\right )+\ln \left (2\right ) {\mathrm e}^{-1} \left (1-\frac {\left (6 x +2\right ) {\mathrm e}^{-3 x}}{2}\right )+\frac {{\mathrm e}^{-1} \left (6-\frac {\left (108 x^{3}+108 x^{2}+72 x +24\right ) {\mathrm e}^{-3 x}}{4}\right )}{108}-\frac {{\mathrm e}^{-1} \left (2-\frac {\left (27 x^{2}+18 x +6\right ) {\mathrm e}^{-3 x}}{3}\right )}{36}\) | \(83\) |
derivativedivides | \(\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )}{36}+\frac {{\mathrm e}^{-3 x -1}}{108}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{2}}{36}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{3}}{108}+2 \,{\mathrm e}^{-3 x -1} \ln \left (2\right )+\ln \left (2\right ) \left ({\mathrm e}^{-3 x -1} \left (-3 x -1\right )-{\mathrm e}^{-3 x -1}\right )\) | \(87\) |
default | \(\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )}{36}+\frac {{\mathrm e}^{-3 x -1}}{108}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{2}}{36}+\frac {{\mathrm e}^{-3 x -1} \left (-3 x -1\right )^{3}}{108}+2 \,{\mathrm e}^{-3 x -1} \ln \left (2\right )+\ln \left (2\right ) \left ({\mathrm e}^{-3 x -1} \left (-3 x -1\right )-{\mathrm e}^{-3 x -1}\right )\) | \(87\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=-\frac {1}{4} \, {\left (x^{3} + 12 \, x \log \left (2\right )\right )} e^{\left (-3 \, x - 1\right )} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=\frac {\left (- x^{3} - 12 x \log {\left (2 \right )}\right ) e^{- 3 x - 1}}{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (17) = 34\).
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=-{\left (3 \, x + 1\right )} e^{\left (-3 \, x - 1\right )} \log \left (2\right ) - \frac {1}{36} \, {\left (9 \, x^{3} + 9 \, x^{2} + 6 \, x + 2\right )} e^{\left (-3 \, x - 1\right )} + \frac {1}{36} \, {\left (9 \, x^{2} + 6 \, x + 2\right )} e^{\left (-3 \, x - 1\right )} + e^{\left (-3 \, x - 1\right )} \log \left (2\right ) \]
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=-\frac {1}{4} \, {\left (x^{3} + 12 \, x \log \left (2\right )\right )} e^{\left (-3 \, x - 1\right )} \]
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Time = 8.74 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {1}{4} e^{-1-3 x} \left (-3 x^2+3 x^3+(4-12 x) \log (2)+(-4+12 x) \log (16)\right ) \, dx=-\frac {x\,{\mathrm {e}}^{-3\,x-1}\,\left (x^2+12\,\ln \left (2\right )\right )}{4} \]
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