Integrand size = 77, antiderivative size = 24 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=7500 \left (e^{x^2}+\frac {e^{-2+e^x-x}}{x}\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(24)=48\).
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.62, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {14, 2240, 2326} \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=\frac {7500 e^{-2 \left (x-e^x+2\right )} \left (x-e^x x\right )}{\left (1-e^x\right ) x^3}+7500 e^{2 x^2}+\frac {15000 e^{x^2-x+e^x-2} \left (-2 x^2-e^x x+x\right )}{\left (-2 x-e^x+1\right ) x^2} \]
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Rule 14
Rule 2240
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \int \left (30000 e^{2 x^2} x+\frac {15000 e^{2 \left (-2+e^x-x\right )} \left (-1-x+e^x x\right )}{x^3}+\frac {15000 e^{-2+e^x-x+x^2} \left (-1-x+e^x x+2 x^2\right )}{x^2}\right ) \, dx \\ & = 15000 \int \frac {e^{2 \left (-2+e^x-x\right )} \left (-1-x+e^x x\right )}{x^3} \, dx+15000 \int \frac {e^{-2+e^x-x+x^2} \left (-1-x+e^x x+2 x^2\right )}{x^2} \, dx+30000 \int e^{2 x^2} x \, dx \\ & = 7500 e^{2 x^2}+\frac {7500 e^{-2 \left (2-e^x+x\right )} \left (x-e^x x\right )}{\left (1-e^x\right ) x^3}+\frac {15000 e^{-2+e^x-x+x^2} \left (x-e^x x-2 x^2\right )}{\left (1-e^x-2 x\right ) x^2} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=7500 \left (e^{2 x^2}+\frac {e^{-4+2 e^x-2 x}}{x^2}\right )+\frac {15000 e^{-2+e^x-x+x^2}}{x} \]
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Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71
method | result | size |
risch | \(7500 \,{\mathrm e}^{2 x^{2}}+\frac {7500 \,{\mathrm e}^{2 \,{\mathrm e}^{x}-4-2 x}}{x^{2}}+\frac {15000 \,{\mathrm e}^{x^{2}+{\mathrm e}^{x}-2-x}}{x}\) | \(41\) |
parallelrisch | \(\frac {7500 \,{\mathrm e}^{2 x^{2}} x^{2}+15000 \,{\mathrm e}^{x^{2}} x \,{\mathrm e}^{{\mathrm e}^{x}-2-x}+7500 \,{\mathrm e}^{2 \,{\mathrm e}^{x}-4-2 x}}{x^{2}}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=\frac {7500 \, {\left (x^{2} e^{\left (4 \, x^{2}\right )} + 2 \, x e^{\left (3 \, x^{2} - x + e^{x} - 2\right )} + e^{\left (2 \, x^{2} - 2 \, x + 2 \, e^{x} - 4\right )}\right )} e^{\left (-2 \, x^{2}\right )}}{x^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=7500 e^{2 x^{2}} + \frac {15000 x^{2} e^{x^{2}} e^{- x + e^{x} - 2} + 7500 x e^{- 2 x + 2 e^{x} - 4}}{x^{3}} \]
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none
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=\frac {7500 \, {\left (2 \, x e^{\left (x^{2} + x + e^{x} + 2\right )} + e^{\left (2 \, e^{x}\right )}\right )} e^{\left (-2 \, x - 4\right )}}{x^{2}} + 7500 \, e^{\left (2 \, x^{2}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=\frac {7500 \, {\left (x^{2} e^{\left (2 \, x^{2} + x + 4\right )} + 2 \, x e^{\left (x^{2} + e^{x} + 2\right )} + e^{\left (-x + 2 \, e^{x}\right )}\right )} e^{\left (-x - 4\right )}}{x^{2}} \]
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Time = 0.65 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {30000 e^{2 x^2} x^4+e^{-4+2 e^x-2 x} \left (-15000-15000 x+15000 e^x x\right )+e^{-2+e^x-x+x^2} \left (-15000 x-15000 x^2+15000 e^x x^2+30000 x^3\right )}{x^3} \, dx=7500\,{\mathrm {e}}^{2\,x^2}+\frac {7500\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{x^2}+\frac {15000\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-2}}{x} \]
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