Integrand size = 115, antiderivative size = 24 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=e^{\frac {9 x}{\left (-3+e^e\right ) (4 x+x \log (2))^2}} x \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6, 2326} \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=x e^{-\frac {9}{\left (3-e^e\right ) x (4+\log (2))^2}} \]
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Rule 6
Rule 2326
Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}\right ) \left (-9+x (-48-24 \log (2))-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx \\ & = \int \frac {\exp \left (\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}\right ) \left (-9+x \left (-48-24 \log (2)-3 \log ^2(2)\right )+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx \\ & = \int \frac {\exp \left (\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}\right ) \left (-9+x \left (-48-24 \log (2)-3 \log ^2(2)\right )+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{x (-48-24 \log (2))-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx \\ & = \int \frac {\exp \left (\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}\right ) \left (-9+x \left (-48-24 \log (2)-3 \log ^2(2)\right )+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{x \left (-48-24 \log (2)-3 \log ^2(2)\right )+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx \\ & = e^{-\frac {9}{\left (3-e^e\right ) x (4+\log (2))^2}} x \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=e^{\frac {9}{\left (-3+e^e\right ) x (4+\log (2))^2}} x \]
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Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
method | result | size |
risch | \(x \,{\mathrm e}^{\frac {9}{x \left (4+\ln \left (2\right )\right )^{2} \left ({\mathrm e}^{{\mathrm e}}-3\right )}}\) | \(22\) |
gosper | \(x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}\) | \(43\) |
norman | \(x \,{\mathrm e}^{\frac {9}{\left (x \ln \left (2\right )^{2}+8 x \ln \left (2\right )+16 x \right ) {\mathrm e}^{{\mathrm e}}-3 x \ln \left (2\right )^{2}-24 x \ln \left (2\right )-48 x}}\) | \(43\) |
parallelrisch | \(\frac {{\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2} x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right ) x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}-3 \ln \left (2\right )^{2} x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}+16 \,{\mathrm e}^{{\mathrm e}} x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}-24 \ln \left (2\right ) x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}-48 x \,{\mathrm e}^{\frac {9}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48\right )}}}{{\mathrm e}^{{\mathrm e}} \ln \left (2\right )^{2}+8 \,{\mathrm e}^{{\mathrm e}} \ln \left (2\right )-3 \ln \left (2\right )^{2}+16 \,{\mathrm e}^{{\mathrm e}}-24 \ln \left (2\right )-48}\) | \(315\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1029\) |
default | \(\text {Expression too large to display}\) | \(1029\) |
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=x e^{\left (-\frac {9}{3 \, x \log \left (2\right )^{2} - {\left (x \log \left (2\right )^{2} + 8 \, x \log \left (2\right ) + 16 \, x\right )} e^{e} + 24 \, x \log \left (2\right ) + 48 \, x}\right )} \]
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Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=x e^{\frac {9}{- 48 x - 24 x \log {\left (2 \right )} - 3 x \log {\left (2 \right )}^{2} + \left (x \log {\left (2 \right )}^{2} + 8 x \log {\left (2 \right )} + 16 x\right ) e^{e}}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.46 (sec) , antiderivative size = 266, normalized size of antiderivative = 11.08 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=\frac {9 \, {\rm Ei}\left (\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right )}{{\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}} - \frac {9 \, e^{e} \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right ) \log \left (2\right )^{2}}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} - \frac {72 \, e^{e} \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right ) \log \left (2\right )}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} + \frac {27 \, \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right ) \log \left (2\right )^{2}}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} - \frac {144 \, e^{e} \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right )}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} + \frac {216 \, \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right ) \log \left (2\right )}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} + \frac {432 \, \Gamma \left (-1, -\frac {9}{x {\left (e^{e} - 3\right )} {\left (\log \left (2\right ) + 4\right )}^{2}}\right )}{{\left (e^{e} - 3\right )}^{2} {\left (\log \left (2\right ) + 4\right )}^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (23) = 46\).
Time = 0.48 (sec) , antiderivative size = 806, normalized size of antiderivative = 33.58 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=\text {Too large to display} \]
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Time = 12.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96 \[ \int \frac {e^{\frac {9}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )}} \left (-9-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )\right )}{-48 x-24 x \log (2)-3 x \log ^2(2)+e^e \left (16 x+8 x \log (2)+x \log ^2(2)\right )} \, dx=x\,{\mathrm {e}}^{-\frac {9}{48\,x+24\,x\,\ln \left (2\right )-16\,x\,{\mathrm {e}}^{\mathrm {e}}+3\,x\,{\ln \left (2\right )}^2-8\,x\,{\mathrm {e}}^{\mathrm {e}}\,\ln \left (2\right )-x\,{\mathrm {e}}^{\mathrm {e}}\,{\ln \left (2\right )}^2}} \]
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