Integrand size = 151, antiderivative size = 25 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {e^x}{\left (\frac {\log (x)}{9}+x (-4-x+x \log (x))\right )^2} \]
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\[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=\int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {81 e^x \left (-2+72 x-18 x^2-9 x^3+x \left (1-36 x+9 x^2\right ) \log (x)\right )}{x \left (9 x (4+x)-\left (1+9 x^2\right ) \log (x)\right )^3} \, dx \\ & = 81 \int \frac {e^x \left (-2+72 x-18 x^2-9 x^3+x \left (1-36 x+9 x^2\right ) \log (x)\right )}{x \left (9 x (4+x)-\left (1+9 x^2\right ) \log (x)\right )^3} \, dx \\ & = 81 \int \left (\frac {2 e^x \left (1-36 x+324 x^3+81 x^4\right )}{x \left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x \left (-1+36 x-9 x^2\right )}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx \\ & = 81 \int \frac {e^x \left (-1+36 x-9 x^2\right )}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+162 \int \frac {e^x \left (1-36 x+324 x^3+81 x^4\right )}{x \left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx \\ & = 81 \int \left (-\frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}+\frac {36 e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx+162 \int \left (\frac {36 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {9 e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}-\frac {18 e^x (4+x)}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx \\ & = -\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \frac {e^x (4+x)}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+2916 \int \frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx \\ & = -\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \left (\frac {4 e^x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx+2916 \int \left (-\frac {e^x}{6 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}+\frac {e^x}{6 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2}\right ) \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx \\ & = -\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \frac {e^x x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-11664 \int \frac {e^x}{\left (1+9 x^2\right ) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx \\ & = -\left (81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx\right )+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-2916 \int \left (-\frac {e^x}{6 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {e^x}{6 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-11664 \int \left (\frac {i e^x}{2 (i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}+\frac {i e^x}{2 (i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3}\right ) \, dx \\ & = -\left (5832 i \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx\right )-5832 i \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-81 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+162 \int \frac {e^x}{x \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx-486 \int \frac {e^x}{(i-3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+486 \int \frac {e^x}{(i+3 x) \left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \, dx+1458 \int \frac {e^x x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx+5832 \int \frac {e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^3} \, dx \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 e^x}{\left (-36 x-9 x^2+\log (x)+9 x^2 \log (x)\right )^2} \]
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Time = 3.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {81 \,{\mathrm e}^{x}}{\left (9 x^{2} \ln \left (x \right )-9 x^{2}+\ln \left (x \right )-36 x \right )^{2}}\) | \(25\) |
parallelrisch | \(-\frac {81 \,{\mathrm e}^{x}}{81 x^{4} \ln \left (x \right )^{2}-162 x^{4} \ln \left (x \right )+81 x^{4}-648 x^{3} \ln \left (x \right )+18 x^{2} \ln \left (x \right )^{2}+648 x^{3}-18 x^{2} \ln \left (x \right )+1296 x^{2}-72 x \ln \left (x \right )+\ln \left (x \right )^{2}}\) | \(71\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.12 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=- \frac {81 e^{x}}{81 x^{4} \log {\left (x \right )}^{2} - 162 x^{4} \log {\left (x \right )} + 81 x^{4} - 648 x^{3} \log {\left (x \right )} + 648 x^{3} + 18 x^{2} \log {\left (x \right )}^{2} - 18 x^{2} \log {\left (x \right )} + 1296 x^{2} - 72 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} + 648 \, x^{3} + {\left (81 \, x^{4} + 18 \, x^{2} + 1\right )} \log \left (x\right )^{2} + 1296 \, x^{2} - 18 \, {\left (9 \, x^{4} + 36 \, x^{3} + x^{2} + 4 \, x\right )} \log \left (x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.80 \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=-\frac {81 \, e^{x}}{81 \, x^{4} \log \left (x\right )^{2} - 162 \, x^{4} \log \left (x\right ) + 81 \, x^{4} - 648 \, x^{3} \log \left (x\right ) + 18 \, x^{2} \log \left (x\right )^{2} + 648 \, x^{3} - 18 \, x^{2} \log \left (x\right ) + 1296 \, x^{2} - 72 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \]
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Timed out. \[ \int \frac {e^x \left (162-5832 x+1458 x^2+729 x^3\right )+e^x \left (-81 x+2916 x^2-729 x^3\right ) \log (x)}{-46656 x^4-34992 x^5-8748 x^6-729 x^7+\left (3888 x^3+1944 x^4+35235 x^5+17496 x^6+2187 x^7\right ) \log (x)+\left (-108 x^2-27 x^3-1944 x^4-486 x^5-8748 x^6-2187 x^7\right ) \log ^2(x)+\left (x+27 x^3+243 x^5+729 x^7\right ) \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^x\,\left (729\,x^3+1458\,x^2-5832\,x+162\right )-{\mathrm {e}}^x\,\ln \left (x\right )\,\left (729\,x^3-2916\,x^2+81\,x\right )}{{\ln \left (x\right )}^2\,\left (2187\,x^7+8748\,x^6+486\,x^5+1944\,x^4+27\,x^3+108\,x^2\right )-{\ln \left (x\right )}^3\,\left (729\,x^7+243\,x^5+27\,x^3+x\right )-\ln \left (x\right )\,\left (2187\,x^7+17496\,x^6+35235\,x^5+1944\,x^4+3888\,x^3\right )+46656\,x^4+34992\,x^5+8748\,x^6+729\,x^7} \,d x \]
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