Integrand size = 66, antiderivative size = 32 \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {1}{4} e^{-\frac {e^x}{2}-x \left (1-e^x x\right ) \log (3)} (-5+4 x) \]
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Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(32)=64\).
Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.88, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {12, 2326} \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {3^{e^x x^2-x} e^{-\frac {e^x}{2}} \left (e^x \left (-2 \left (-4 x^3-3 x^2+10 x\right ) \log (3)-4 x+5\right )+2 (5-4 x) \log (3)\right )}{4 \left (-e^x \left (1-2 x^2 \log (3)\right )+4 e^x x \log (3)-\log (9)\right )} \]
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Rule 12
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \int \exp \left (\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )\right ) \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx \\ & = \frac {3^{-x+e^x x^2} e^{-\frac {e^x}{2}} \left (2 (5-4 x) \log (3)+e^x \left (5-4 x-2 \left (10 x-3 x^2-4 x^3\right ) \log (3)\right )\right )}{4 \left (4 e^x x \log (3)-e^x \left (1-2 x^2 \log (3)\right )-\log (9)\right )} \\ \end{align*}
\[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx \]
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Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\left (x -\frac {5}{4}\right ) {\mathrm e}^{-\frac {\left (-2 x^{2} \ln \left (3\right )+1\right ) {\mathrm e}^{x}}{2}-x \ln \left (3\right )}\) | \(26\) |
risch | \(\frac {\left (8 x -10\right ) 3^{{\mathrm e}^{x} x^{2}} 3^{-x} {\mathrm e}^{-\frac {{\mathrm e}^{x}}{2}}}{8}\) | \(28\) |
parallelrisch | \(\frac {\left (16 x -20\right ) {\mathrm e}^{x^{2} \ln \left (3\right ) {\mathrm e}^{x}-\frac {{\mathrm e}^{x}}{2}-x \ln \left (3\right )}}{16}\) | \(29\) |
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Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {1}{4} \, {\left (4 \, x - 5\right )} e^{\left (\frac {1}{2} \, {\left (2 \, x^{2} \log \left (3\right ) - 1\right )} e^{x} - x \log \left (3\right )\right )} \]
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Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {\left (4 x - 5\right ) e^{- x \log {\left (3 \right )} - \left (- x^{2} \log {\left (3 \right )} + \frac {1}{2}\right ) e^{x}}}{4} \]
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Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {1}{4} \, {\left (4 \, x - 5\right )} e^{\left (x^{2} e^{x} \log \left (3\right ) - x \log \left (3\right ) - \frac {1}{2} \, e^{x}\right )} \]
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\[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\int { \frac {1}{8} \, {\left ({\left (2 \, {\left (4 \, x^{3} + 3 \, x^{2} - 10 \, x\right )} \log \left (3\right ) - 4 \, x + 5\right )} e^{x} - 2 \, {\left (4 \, x - 5\right )} \log \left (3\right ) + 8\right )} e^{\left (\frac {1}{2} \, {\left (2 \, x^{2} \log \left (3\right ) - 1\right )} e^{x} - x \log \left (3\right )\right )} \,d x } \]
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Time = 11.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {1}{8} e^{\frac {1}{2} \left (-2 x \log (3)-e^x \left (1-2 x^2 \log (3)\right )\right )} \left (8+(10-8 x) \log (3)+e^x \left (5-4 x+\left (-20 x+6 x^2+8 x^3\right ) \log (3)\right )\right ) \, dx=\frac {3^{x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x}{2}}\,\left (4\,x-5\right )}{4\,3^x} \]
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