Integrand size = 81, antiderivative size = 20 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=3+\left (-3+e^{e^x}\right )^2 \log ^2(4+2 x) \]
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\[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=\int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (3-e^{e^x}\right ) \left (3-e^{e^x}-e^{e^x+x} (2+x) \log (2 (2+x))\right ) \log (4+2 x)}{2+x} \, dx \\ & = 2 \int \frac {\left (3-e^{e^x}\right ) \left (3-e^{e^x}-e^{e^x+x} (2+x) \log (2 (2+x))\right ) \log (4+2 x)}{2+x} \, dx \\ & = 2 \int \left (\frac {\left (3-e^{e^x}\right )^2 \log (4+2 x)}{2+x}+e^{e^x+x} \left (-3+e^{e^x}\right ) \log ^2(4+2 x)\right ) \, dx \\ & = 2 \int \frac {\left (3-e^{e^x}\right )^2 \log (4+2 x)}{2+x} \, dx+2 \int e^{e^x+x} \left (-3+e^{e^x}\right ) \log ^2(4+2 x) \, dx \\ & = 2 \int \left (\frac {6 e^{e^x} \log (4+2 x)}{-2-x}+\frac {9 \log (4+2 x)}{2+x}+\frac {e^{2 e^x} \log (4+2 x)}{2+x}\right ) \, dx+2 \int \left (-3 e^{e^x+x} \log ^2(4+2 x)+e^{2 e^x+x} \log ^2(4+2 x)\right ) \, dx \\ & = 2 \int \frac {e^{2 e^x} \log (4+2 x)}{2+x} \, dx+2 \int e^{2 e^x+x} \log ^2(4+2 x) \, dx-6 \int e^{e^x+x} \log ^2(4+2 x) \, dx+12 \int \frac {e^{e^x} \log (4+2 x)}{-2-x} \, dx+18 \int \frac {\log (4+2 x)}{2+x} \, dx \\ & = 2 \int e^{2 e^x+x} \log ^2(4+2 x) \, dx-2 \int \frac {\int \frac {e^{2 e^x}}{2+x} \, dx}{2+x} \, dx-6 \int e^{e^x+x} \log ^2(4+2 x) \, dx+9 \text {Subst}\left (\int \frac {2 \log (x)}{x} \, dx,x,4+2 x\right )-12 \int \frac {\int -\frac {e^{e^x}}{2+x} \, dx}{2+x} \, dx+(2 \log (4+2 x)) \int \frac {e^{2 e^x}}{2+x} \, dx+(12 \log (4+2 x)) \int \frac {e^{e^x}}{-2-x} \, dx \\ & = 2 \int e^{2 e^x+x} \log ^2(4+2 x) \, dx-2 \int \frac {\int \frac {e^{2 e^x}}{2+x} \, dx}{2+x} \, dx-6 \int e^{e^x+x} \log ^2(4+2 x) \, dx+12 \int \frac {\int \frac {e^{e^x}}{2+x} \, dx}{2+x} \, dx+18 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,4+2 x\right )+(2 \log (4+2 x)) \int \frac {e^{2 e^x}}{2+x} \, dx+(12 \log (4+2 x)) \int \frac {e^{e^x}}{-2-x} \, dx \\ & = 9 \log ^2(2 (2+x))+2 \int e^{2 e^x+x} \log ^2(4+2 x) \, dx-2 \int \frac {\int \frac {e^{2 e^x}}{2+x} \, dx}{2+x} \, dx-6 \int e^{e^x+x} \log ^2(4+2 x) \, dx+12 \int \frac {\int \frac {e^{e^x}}{2+x} \, dx}{2+x} \, dx+(2 \log (4+2 x)) \int \frac {e^{2 e^x}}{2+x} \, dx+(12 \log (4+2 x)) \int \frac {e^{e^x}}{-2-x} \, dx \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=\left (-3+e^{e^x}\right )^2 \log ^2(2 (2+x)) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(18)=36\).
Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95
method | result | size |
risch | \(\ln \left (4+2 x \right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}}-6 \ln \left (4+2 x \right )^{2} {\mathrm e}^{{\mathrm e}^{x}}+9 \ln \left (4+2 x \right )^{2}\) | \(39\) |
parallelrisch | \(\ln \left (4+2 x \right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}}-6 \ln \left (4+2 x \right )^{2} {\mathrm e}^{{\mathrm e}^{x}}+9 \ln \left (4+2 x \right )^{2}\) | \(39\) |
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Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=e^{\left (2 \, e^{x}\right )} \log \left (2 \, x + 4\right )^{2} - 6 \, e^{\left (e^{x}\right )} \log \left (2 \, x + 4\right )^{2} + 9 \, \log \left (2 \, x + 4\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 14.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=e^{2 e^{x}} \log {\left (2 x + 4 \right )}^{2} - 6 e^{e^{x}} \log {\left (2 x + 4 \right )}^{2} + 9 \log {\left (2 x + 4 \right )}^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (18) = 36\).
Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.30 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx={\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}\right )} e^{\left (2 \, e^{x}\right )} - 6 \, {\left (\log \left (2\right )^{2} + 2 \, \log \left (2\right ) \log \left (x + 2\right ) + \log \left (x + 2\right )^{2}\right )} e^{\left (e^{x}\right )} + 18 \, \log \left (2\right ) \log \left (x + 2\right ) + 9 \, \log \left (x + 2\right )^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.15 \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=e^{\left (2 \, e^{x}\right )} \log \left (2\right )^{2} - 6 \, e^{\left (e^{x}\right )} \log \left (2\right )^{2} + 2 \, e^{\left (2 \, e^{x}\right )} \log \left (2\right ) \log \left (x + 2\right ) - 12 \, e^{\left (e^{x}\right )} \log \left (2\right ) \log \left (x + 2\right ) + e^{\left (2 \, e^{x}\right )} \log \left (x + 2\right )^{2} - 6 \, e^{\left (e^{x}\right )} \log \left (x + 2\right )^{2} + 18 \, \log \left (2\right ) \log \left (x + 2\right ) + 9 \, \log \left (x + 2\right )^{2} \]
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Timed out. \[ \int \frac {18 \log (4+2 x)+e^{e^x} \left (-12 \log (4+2 x)+e^x (-12-6 x) \log ^2(4+2 x)\right )+e^{2 e^x} \left (2 \log (4+2 x)+e^x (4+2 x) \log ^2(4+2 x)\right )}{2+x} \, dx=\int \frac {18\,\ln \left (2\,x+4\right )-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x\,\left (6\,x+12\right )\,{\ln \left (2\,x+4\right )}^2+12\,\ln \left (2\,x+4\right )\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^x\,\left (2\,x+4\right )\,{\ln \left (2\,x+4\right )}^2+2\,\ln \left (2\,x+4\right )\right )}{x+2} \,d x \]
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