Integrand size = 196, antiderivative size = 29 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \]
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\[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {8 \left (e^x-x \left (2+x^3\right )\right ) \log ^2(3) \log (x) \left (-\left ((-3+x) \left (e^x-x \left (2+x^3\right )\right )\right )-x \left (6+e^x (-4+x)+12 x^3-3 x^4\right ) \log (x)\right )}{(3-x)^3 x} \, dx \\ & = \left (8 \log ^2(3)\right ) \int \frac {\left (e^x-x \left (2+x^3\right )\right ) \log (x) \left (-\left ((-3+x) \left (e^x-x \left (2+x^3\right )\right )\right )-x \left (6+e^x (-4+x)+12 x^3-3 x^4\right ) \log (x)\right )}{(3-x)^3 x} \, dx \\ & = \left (8 \log ^2(3)\right ) \int \left (\frac {x \left (2+x^3\right )^2 \log (x)}{(-3+x)^2}-\frac {6 x \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}-\frac {12 x^4 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}+\frac {3 x^5 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3}+\frac {e^{2 x} \log (x) \left (-3+x-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^3 x}-\frac {e^x \log (x) \left (-12+4 x-6 x^3+2 x^4-6 \log (x)-8 x \log (x)+2 x^2 \log (x)-12 x^3 \log (x)-x^4 \log (x)+x^5 \log (x)\right )}{(-3+x)^3}\right ) \, dx \\ & = \left (8 \log ^2(3)\right ) \int \frac {x \left (2+x^3\right )^2 \log (x)}{(-3+x)^2} \, dx+\left (8 \log ^2(3)\right ) \int \frac {e^{2 x} \log (x) \left (-3+x-4 x \log (x)+x^2 \log (x)\right )}{(-3+x)^3 x} \, dx-\left (8 \log ^2(3)\right ) \int \frac {e^x \log (x) \left (-12+4 x-6 x^3+2 x^4-6 \log (x)-8 x \log (x)+2 x^2 \log (x)-12 x^3 \log (x)-x^4 \log (x)+x^5 \log (x)\right )}{(-3+x)^3} \, dx+\left (24 \log ^2(3)\right ) \int \frac {x^5 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx-\left (48 \log ^2(3)\right ) \int \frac {x \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx-\left (96 \log ^2(3)\right ) \int \frac {x^4 \left (2+x^3\right ) \log ^2(x)}{(-3+x)^3} \, dx \\ & = \left (8 \log ^2(3)\right ) \int \left (1566 \log (x)+\frac {2523 \log (x)}{(-3+x)^2}+\frac {5539 \log (x)}{-3+x}+429 x \log (x)+112 x^2 \log (x)+27 x^3 \log (x)+6 x^4 \log (x)+x^5 \log (x)\right ) \, dx+\left (8 \log ^2(3)\right ) \int \left (\frac {e^{2 x} \log (x)}{(-3+x)^2 x}+\frac {e^{2 x} (-4+x) \log ^2(x)}{(-3+x)^3}\right ) \, dx-\left (8 \log ^2(3)\right ) \int \left (\frac {2 e^x \left (2+x^3\right ) \log (x)}{(-3+x)^2}+\frac {e^x \left (-6-8 x+2 x^2-12 x^3-x^4+x^5\right ) \log ^2(x)}{(-3+x)^3}\right ) \, dx+\left (24 \log ^2(3)\right ) \int \left (5211 \log ^2(x)+\frac {7047 \log ^2(x)}{(-3+x)^3}+\frac {18306 \log ^2(x)}{(-3+x)^2}+\frac {20952 \log ^2(x)}{-3+x}+1233 x \log ^2(x)+272 x^2 \log ^2(x)+54 x^3 \log ^2(x)+9 x^4 \log ^2(x)+x^5 \log ^2(x)\right ) \, dx-\left (48 \log ^2(3)\right ) \int \left (9 \log ^2(x)+\frac {87 \log ^2(x)}{(-3+x)^3}+\frac {110 \log ^2(x)}{(-3+x)^2}+\frac {54 \log ^2(x)}{-3+x}+x \log ^2(x)\right ) \, dx-\left (96 \log ^2(3)\right ) \int \left (1233 \log ^2(x)+\frac {2349 \log ^2(x)}{(-3+x)^3}+\frac {5319 \log ^2(x)}{(-3+x)^2}+\frac {5211 \log ^2(x)}{-3+x}+272 x \log ^2(x)+54 x^2 \log ^2(x)+9 x^3 \log ^2(x)+x^4 \log ^2(x)\right ) \, dx \\ & = \left (8 \log ^2(3)\right ) \int \frac {e^{2 x} \log (x)}{(-3+x)^2 x} \, dx+\left (8 \log ^2(3)\right ) \int x^5 \log (x) \, dx+\left (8 \log ^2(3)\right ) \int \frac {e^{2 x} (-4+x) \log ^2(x)}{(-3+x)^3} \, dx-\left (8 \log ^2(3)\right ) \int \frac {e^x \left (-6-8 x+2 x^2-12 x^3-x^4+x^5\right ) \log ^2(x)}{(-3+x)^3} \, dx-\left (16 \log ^2(3)\right ) \int \frac {e^x \left (2+x^3\right ) \log (x)}{(-3+x)^2} \, dx+\left (24 \log ^2(3)\right ) \int x^5 \log ^2(x) \, dx+\left (48 \log ^2(3)\right ) \int x^4 \log (x) \, dx-\left (48 \log ^2(3)\right ) \int x \log ^2(x) \, dx-\left (96 \log ^2(3)\right ) \int x^4 \log ^2(x) \, dx+\left (216 \log ^2(3)\right ) \int x^3 \log (x) \, dx+\left (216 \log ^2(3)\right ) \int x^4 \log ^2(x) \, dx-\left (432 \log ^2(3)\right ) \int \log ^2(x) \, dx-\left (864 \log ^2(3)\right ) \int x^3 \log ^2(x) \, dx+\left (896 \log ^2(3)\right ) \int x^2 \log (x) \, dx+\left (1296 \log ^2(3)\right ) \int x^3 \log ^2(x) \, dx-\left (2592 \log ^2(3)\right ) \int \frac {\log ^2(x)}{-3+x} \, dx+\left (3432 \log ^2(3)\right ) \int x \log (x) \, dx-\left (4176 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^3} \, dx-\left (5184 \log ^2(3)\right ) \int x^2 \log ^2(x) \, dx-\left (5280 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^2} \, dx+\left (6528 \log ^2(3)\right ) \int x^2 \log ^2(x) \, dx+\left (12528 \log ^2(3)\right ) \int \log (x) \, dx+\left (20184 \log ^2(3)\right ) \int \frac {\log (x)}{(-3+x)^2} \, dx-\left (26112 \log ^2(3)\right ) \int x \log ^2(x) \, dx+\left (29592 \log ^2(3)\right ) \int x \log ^2(x) \, dx+\left (44312 \log ^2(3)\right ) \int \frac {\log (x)}{-3+x} \, dx-\left (118368 \log ^2(3)\right ) \int \log ^2(x) \, dx+\left (125064 \log ^2(3)\right ) \int \log ^2(x) \, dx+\left (169128 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^3} \, dx-\left (225504 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^3} \, dx+\left (439344 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^2} \, dx-\left (500256 \log ^2(3)\right ) \int \frac {\log ^2(x)}{-3+x} \, dx+\left (502848 \log ^2(3)\right ) \int \frac {\log ^2(x)}{-3+x} \, dx-\left (510624 \log ^2(3)\right ) \int \frac {\log ^2(x)}{(-3+x)^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 5.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \left (-e^x+2 x+x^4\right )^2 \log ^2(3) \log ^2(x)}{(-3+x)^2} \]
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Time = 4.60 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59
method | result | size |
risch | \(\frac {4 \left (x^{8}+4 x^{5}-2 \,{\mathrm e}^{x} x^{4}+4 x^{2}-4 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}\right ) \ln \left (3\right )^{2} \ln \left (x \right )^{2}}{\left (-3+x \right )^{2}}\) | \(46\) |
parallelrisch | \(\frac {4 \ln \left (x \right )^{2} \ln \left (3\right )^{2} x^{8}+16 \ln \left (x \right )^{2} \ln \left (3\right )^{2} x^{5}-8 \ln \left (x \right )^{2} {\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+16 x^{2} \ln \left (3\right )^{2} \ln \left (x \right )^{2}-16 \ln \left (x \right )^{2} {\mathrm e}^{x} \ln \left (3\right )^{2} x +4 \ln \left (3\right )^{2} \ln \left (x \right )^{2} {\mathrm e}^{2 x}}{x^{2}-6 x +9}\) | \(94\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=-\frac {4 \, {\left (2 \, {\left (x^{4} + 2 \, x\right )} e^{x} \log \left (3\right )^{2} - {\left (x^{8} + 4 \, x^{5} + 4 \, x^{2}\right )} \log \left (3\right )^{2} - e^{\left (2 \, x\right )} \log \left (3\right )^{2}\right )} \log \left (x\right )^{2}}{x^{2} - 6 \, x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (27) = 54\).
Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 6.93 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {\left (4 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 24 x \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 36 \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}\right ) e^{2 x} + \left (- 8 x^{6} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 48 x^{5} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 72 x^{4} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 16 x^{3} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} + 96 x^{2} \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2} - 144 x \log {\left (3 \right )}^{2} \log {\left (x \right )}^{2}\right ) e^{x}}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} + \frac {\left (4 x^{8} \log {\left (3 \right )}^{2} + 16 x^{5} \log {\left (3 \right )}^{2} + 16 x^{2} \log {\left (3 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{2} - 6 x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (28) = 56\).
Time = 0.34 (sec) , antiderivative size = 160, normalized size of antiderivative = 5.52 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=16 \, {\left (\frac {3 \, {\left (2 \, x - 3\right )} \log \left (x\right )}{x^{2} - 6 \, x + 9} + \frac {3}{x - 3} - \log \left (x - 3\right ) + \log \left (x\right )\right )} \log \left (3\right )^{2} + 16 \, \log \left (3\right )^{2} \log \left (x - 3\right ) - \frac {4 \, {\left (4 \, x^{2} \log \left (3\right )^{2} \log \left (x\right ) - e^{\left (2 \, x\right )} \log \left (3\right )^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{4} \log \left (3\right )^{2} + 2 \, x \log \left (3\right )^{2}\right )} e^{x} \log \left (x\right )^{2} + 12 \, x \log \left (3\right )^{2} - {\left (x^{8} \log \left (3\right )^{2} + 4 \, x^{5} \log \left (3\right )^{2} + 4 \, x^{2} \log \left (3\right )^{2}\right )} \log \left (x\right )^{2} - 36 \, \log \left (3\right )^{2}\right )}}{x^{2} - 6 \, x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).
Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\frac {4 \, {\left (x^{8} \log \left (3\right )^{2} \log \left (x\right )^{2} + 4 \, x^{5} \log \left (3\right )^{2} \log \left (x\right )^{2} - 2 \, x^{4} e^{x} \log \left (3\right )^{2} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (3\right )^{2} \log \left (x\right )^{2} - 4 \, x e^{x} \log \left (3\right )^{2} \log \left (x\right )^{2} + e^{\left (2 \, x\right )} \log \left (3\right )^{2} \log \left (x\right )^{2}\right )}}{x^{2} - 6 \, x + 9} \]
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Timed out. \[ \int \frac {\left (e^{2 x} (-24+8 x) \log ^2(3)+e^x \left (96 x-32 x^2+48 x^4-16 x^5\right ) \log ^2(3)+\left (-96 x^2+32 x^3-96 x^5+32 x^6-24 x^8+8 x^9\right ) \log ^2(3)\right ) \log (x)+\left (e^{2 x} \left (-32 x+8 x^2\right ) \log ^2(3)+e^x \left (48 x+64 x^2-16 x^3+96 x^4+8 x^5-8 x^6\right ) \log ^2(3)+\left (-96 x^2-240 x^5+48 x^6-96 x^8+24 x^9\right ) \log ^2(3)\right ) \log ^2(x)}{-27 x+27 x^2-9 x^3+x^4} \, dx=\int \frac {{\ln \left (x\right )}^2\,\left ({\ln \left (3\right )}^2\,\left (-24\,x^9+96\,x^8-48\,x^6+240\,x^5+96\,x^2\right )-{\mathrm {e}}^x\,{\ln \left (3\right )}^2\,\left (-8\,x^6+8\,x^5+96\,x^4-16\,x^3+64\,x^2+48\,x\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (32\,x-8\,x^2\right )\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,{\ln \left (3\right )}^2\,\left (8\,x-24\right )-{\ln \left (3\right )}^2\,\left (-8\,x^9+24\,x^8-32\,x^6+96\,x^5-32\,x^3+96\,x^2\right )+{\mathrm {e}}^x\,{\ln \left (3\right )}^2\,\left (-16\,x^5+48\,x^4-32\,x^2+96\,x\right )\right )}{-x^4+9\,x^3-27\,x^2+27\,x} \,d x \]
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