Integrand size = 120, antiderivative size = 21 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=x \log \left (2 \left (2+e^x\right )\right ) \left (3+(5+\log (\log (12)))^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(45\) vs. \(2(21)=42\).
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6, 6820, 12, 6874, 2215, 2221, 2317, 2438, 2320, 2439} \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=x \left (28+\log ^2(\log (12))+10 \log (\log (12))\right )^2 \log \left (\frac {e^x}{2}+1\right )+x \log (4) \left (28+\log ^2(\log (12))+10 \log (\log (12))\right )^2 \]
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Rule 6
Rule 12
Rule 2215
Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2439
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+e^x x (784+560 \log (\log (12)))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx \\ & = \int \frac {e^x x \log ^4(\log (12))+e^x x (784+560 \log (\log (12)))+e^x x \left (156 \log ^2(\log (12))+20 \log ^3(\log (12))\right )+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx \\ & = \int \frac {e^x x \left (156 \log ^2(\log (12))+20 \log ^3(\log (12))\right )+e^x x \left (784+560 \log (\log (12))+\log ^4(\log (12))\right )+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx \\ & = \int \frac {e^x x \left (784+560 \log (\log (12))+156 \log ^2(\log (12))+20 \log ^3(\log (12))+\log ^4(\log (12))\right )+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx \\ & = \int \frac {\left (e^x x+\left (2+e^x\right ) \log \left (2 \left (2+e^x\right )\right )\right ) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2}{2+e^x} \, dx \\ & = \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \int \frac {e^x x+\left (2+e^x\right ) \log \left (2 \left (2+e^x\right )\right )}{2+e^x} \, dx \\ & = \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \int \left (x-\frac {2 x}{2+e^x}+\log \left (2 \left (2+e^x\right )\right )\right ) \, dx \\ & = \frac {1}{2} x^2 \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2+\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \int \log \left (2 \left (2+e^x\right )\right ) \, dx-\left (2 \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2\right ) \int \frac {x}{2+e^x} \, dx \\ & = \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \int \frac {e^x x}{2+e^x} \, dx+\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \text {Subst}\left (\int \frac {\log (4+2 x)}{x} \, dx,x,e^x\right ) \\ & = x \log (4) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2+x \log \left (1+\frac {e^x}{2}\right ) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2-\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \int \log \left (1+\frac {e^x}{2}\right ) \, dx+\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right ) \\ & = x \log (4) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2+x \log \left (1+\frac {e^x}{2}\right ) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2-\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )-\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,e^x\right ) \\ & = x \log (4) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2+x \log \left (1+\frac {e^x}{2}\right ) \left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=\left (28+10 \log (\log (12))+\log ^2(\log (12))\right )^2 \left (\frac {x^2}{2}+x \log (4)+x \log \left (1+2 e^{-x}\right )-\operatorname {PolyLog}\left (2,-2 e^{-x}\right )-\operatorname {PolyLog}\left (2,-\frac {e^x}{2}\right )\right ) \]
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Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71
method | result | size |
norman | \(\left (\ln \left (\ln \left (12\right )\right )^{4}+20 \ln \left (\ln \left (12\right )\right )^{3}+156 \ln \left (\ln \left (12\right )\right )^{2}+560 \ln \left (\ln \left (12\right )\right )+784\right ) x \ln \left (2 \,{\mathrm e}^{x}+4\right )\) | \(36\) |
risch | \(x \left (\ln \left (\ln \left (3\right )+2 \ln \left (2\right )\right )^{4}+20 \ln \left (\ln \left (3\right )+2 \ln \left (2\right )\right )^{3}+156 \ln \left (\ln \left (3\right )+2 \ln \left (2\right )\right )^{2}+560 \ln \left (\ln \left (3\right )+2 \ln \left (2\right )\right )+784\right ) \ln \left (2 \,{\mathrm e}^{x}+4\right )\) | \(56\) |
parallelrisch | \(\ln \left (\ln \left (12\right )\right )^{4} x \ln \left (2 \,{\mathrm e}^{x}+4\right )+20 \ln \left (\ln \left (12\right )\right )^{3} x \ln \left (2 \,{\mathrm e}^{x}+4\right )+156 \ln \left (\ln \left (12\right )\right )^{2} x \ln \left (2 \,{\mathrm e}^{x}+4\right )+560 \ln \left (\ln \left (12\right )\right ) x \ln \left (2 \,{\mathrm e}^{x}+4\right )+784 x \ln \left (2 \,{\mathrm e}^{x}+4\right )\) | \(69\) |
default | \(784 x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )+\ln \left (\ln \left (12\right )\right )^{4} \left (\operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )+x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+20 \ln \left (\ln \left (12\right )\right )^{3} \left (\operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )+x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+156 \ln \left (\ln \left (12\right )\right )^{2} \left (\operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )+x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+560 \ln \left (\ln \left (12\right )\right ) \left (\operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )+x \ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+784 \left (\ln \left (2 \,{\mathrm e}^{x}+4\right )-\ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right ) \ln \left (-\frac {{\mathrm e}^{x}}{2}\right )+\ln \left (\ln \left (12\right )\right )^{4} \left (\left (\ln \left (2 \,{\mathrm e}^{x}+4\right )-\ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right ) \ln \left (-\frac {{\mathrm e}^{x}}{2}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+\ln \left (\ln \left (12\right )\right )^{3} \left (20 \left (\ln \left (2 \,{\mathrm e}^{x}+4\right )-\ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right ) \ln \left (-\frac {{\mathrm e}^{x}}{2}\right )-20 \operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+\ln \left (\ln \left (12\right )\right )^{2} \left (156 \left (\ln \left (2 \,{\mathrm e}^{x}+4\right )-\ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right ) \ln \left (-\frac {{\mathrm e}^{x}}{2}\right )-156 \operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )+\ln \left (\ln \left (12\right )\right ) \left (560 \left (\ln \left (2 \,{\mathrm e}^{x}+4\right )-\ln \left (\frac {{\mathrm e}^{x}}{2}+1\right )\right ) \ln \left (-\frac {{\mathrm e}^{x}}{2}\right )-560 \operatorname {dilog}\left (\frac {{\mathrm e}^{x}}{2}+1\right )\right )\) | \(286\) |
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.95 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx={\left (x \log \left (\log \left (12\right )\right )^{4} + 20 \, x \log \left (\log \left (12\right )\right )^{3} + 156 \, x \log \left (\log \left (12\right )\right )^{2} + 560 \, x \log \left (\log \left (12\right )\right ) + 784 \, x\right )} \log \left (2 \, e^{x} + 4\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.29 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=\left (x \log {\left (\log {\left (12 \right )} \right )}^{4} + 20 x \log {\left (\log {\left (12 \right )} \right )}^{3} + 156 x \log {\left (\log {\left (12 \right )} \right )}^{2} + 560 x \log {\left (\log {\left (12 \right )} \right )} + 784 x\right ) \log {\left (2 e^{x} + 4 \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.95 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx={\left (\log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{4} + 20 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{3} + 156 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{2} + 560 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right ) + 784\right )} x \log \left (2\right ) + {\left (\log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{4} + 20 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{3} + 156 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right )^{2} + 560 \, \log \left (\log \left (3\right ) + 2 \, \log \left (2\right )\right ) + 784\right )} x \log \left (e^{x} + 2\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (20) = 40\).
Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.76 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=x \log \left (2\right ) \log \left (\log \left (12\right )\right )^{4} + x \log \left (e^{x} + 2\right ) \log \left (\log \left (12\right )\right )^{4} + 20 \, x \log \left (2\right ) \log \left (\log \left (12\right )\right )^{3} + 20 \, x \log \left (e^{x} + 2\right ) \log \left (\log \left (12\right )\right )^{3} + 156 \, x \log \left (2\right ) \log \left (\log \left (12\right )\right )^{2} + 156 \, x \log \left (e^{x} + 2\right ) \log \left (\log \left (12\right )\right )^{2} + 560 \, x \log \left (2\right ) \log \left (\log \left (12\right )\right ) + 560 \, x \log \left (e^{x} + 2\right ) \log \left (\log \left (12\right )\right ) + 784 \, x \log \left (2\right ) + 784 \, x \log \left (e^{x} + 2\right ) \]
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Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {784 e^x x+560 e^x x \log (\log (12))+156 e^x x \log ^2(\log (12))+20 e^x x \log ^3(\log (12))+e^x x \log ^4(\log (12))+\log \left (4+2 e^x\right ) \left (1568+784 e^x+\left (1120+560 e^x\right ) \log (\log (12))+\left (312+156 e^x\right ) \log ^2(\log (12))+\left (40+20 e^x\right ) \log ^3(\log (12))+\left (2+e^x\right ) \log ^4(\log (12))\right )}{2+e^x} \, dx=x\,\left (\ln \left (2\right )+\ln \left ({\mathrm {e}}^x+2\right )\right )\,{\left (10\,\ln \left (\ln \left (12\right )\right )+{\ln \left (\ln \left (12\right )\right )}^2+28\right )}^2 \]
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