Integrand size = 132, antiderivative size = 25 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=e^{-\frac {6}{x}+x}+\left (\frac {1}{-3+x}-x \log (2 x)\right )^2 \]
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {6820, 6838, 697, 2404, 2351, 31, 2341, 2342} \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=x^2 \log ^2(2 x)+e^{x-\frac {6}{x}}+\frac {1}{(3-x)^2}+\frac {2 x \log (2 x)}{3-x} \]
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Rule 31
Rule 697
Rule 2341
Rule 2342
Rule 2351
Rule 2404
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-\frac {6}{x}+x} \left (1+\frac {6}{x^2}\right )-\frac {2 \left (10-6 x+x^2\right )}{(-3+x)^3}+\frac {2 \left (3+9 x-6 x^2+x^3\right ) \log (2 x)}{(-3+x)^2}+2 x \log ^2(2 x)\right ) \, dx \\ & = -\left (2 \int \frac {10-6 x+x^2}{(-3+x)^3} \, dx\right )+2 \int \frac {\left (3+9 x-6 x^2+x^3\right ) \log (2 x)}{(-3+x)^2} \, dx+2 \int x \log ^2(2 x) \, dx+\int e^{-\frac {6}{x}+x} \left (1+\frac {6}{x^2}\right ) \, dx \\ & = e^{-\frac {6}{x}+x}+x^2 \log ^2(2 x)-2 \int \left (\frac {1}{(-3+x)^3}+\frac {1}{-3+x}\right ) \, dx-2 \int x \log (2 x) \, dx+2 \int \left (\frac {3 \log (2 x)}{(-3+x)^2}+x \log (2 x)\right ) \, dx \\ & = e^{-\frac {6}{x}+x}+\frac {1}{(3-x)^2}+\frac {x^2}{2}-2 \log (3-x)-x^2 \log (2 x)+x^2 \log ^2(2 x)+2 \int x \log (2 x) \, dx+6 \int \frac {\log (2 x)}{(-3+x)^2} \, dx \\ & = e^{-\frac {6}{x}+x}+\frac {1}{(3-x)^2}-2 \log (3-x)+\frac {2 x \log (2 x)}{3-x}+x^2 \log ^2(2 x)+2 \int \frac {1}{-3+x} \, dx \\ & = e^{-\frac {6}{x}+x}+\frac {1}{(3-x)^2}+\frac {2 x \log (2 x)}{3-x}+x^2 \log ^2(2 x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(25)=50\).
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=e^{-\frac {6}{x}+x}+\frac {1}{(-3+x)^2}+2 \log (3-x)-2 \log (-3+x)-2 \log (x)+\frac {6 \log (2 x)}{3-x}+x^2 \log ^2(2 x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(24)=48\).
Time = 0.57 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20
method | result | size |
default | \({\mathrm e}^{\frac {x^{2}-6}{x}}+x^{2} \ln \left (2 x \right )^{2}-2 \ln \left (-3+x \right )+\frac {1}{\left (-3+x \right )^{2}}+2 \ln \left (2 x -6\right )-\frac {4 \ln \left (2 x \right ) x}{2 x -6}\) | \(55\) |
parts | \({\mathrm e}^{\frac {x^{2}-6}{x}}+x^{2} \ln \left (2 x \right )^{2}-2 \ln \left (-3+x \right )+\frac {1}{\left (-3+x \right )^{2}}+2 \ln \left (2 x -6\right )-\frac {4 \ln \left (2 x \right ) x}{2 x -6}\) | \(55\) |
risch | \(x^{2} \ln \left (2 x \right )^{2}-\frac {6 \ln \left (2 x \right )}{-3+x}-\frac {2 x^{2} \ln \left (x \right )-{\mathrm e}^{\frac {x^{2}-6}{x}} x^{2}-12 x \ln \left (x \right )+6 \,{\mathrm e}^{\frac {x^{2}-6}{x}} x +18 \ln \left (x \right )-9 \,{\mathrm e}^{\frac {x^{2}-6}{x}}-1}{\left (-3+x \right )^{2}}\) | \(88\) |
parallelrisch | \(\frac {6 x^{4} \ln \left (2 x \right )^{2}-36 x^{3} \ln \left (2 x \right )^{2}+6+54 x^{2} \ln \left (2 x \right )^{2}-6 x^{2} \ln \left (x \right )-6 x^{2} \ln \left (2 x \right )+6 \,{\mathrm e}^{\frac {x^{2}-6}{x}} x^{2}+36 x \ln \left (x \right )-36 \,{\mathrm e}^{\frac {x^{2}-6}{x}} x -54 \ln \left (x \right )+54 \ln \left (2 x \right )+54 \,{\mathrm e}^{\frac {x^{2}-6}{x}}}{6 x^{2}-36 x +54}\) | \(119\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=\frac {{\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} \log \left (2 \, x\right )^{2} + {\left (x^{2} - 6 \, x + 9\right )} e^{\left (\frac {x^{2} - 6}{x}\right )} - 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (2 \, x\right ) + 1}{x^{2} - 6 \, x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=x^{2} \log {\left (2 x \right )}^{2} + e^{\frac {x^{2} - 6}{x}} - 2 \log {\left (x \right )} + \frac {1}{x^{2} - 6 x + 9} - \frac {6 \log {\left (2 x \right )}}{x - 3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (25) = 50\).
Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.88 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=\frac {3 \, {\left (4 \, x - 9\right )}}{x^{2} - 6 \, x + 9} - \frac {6 \, {\left (2 \, x - 3\right )}}{x^{2} - 6 \, x + 9} + \frac {x^{3} \log \left (2\right )^{2} - 3 \, x^{2} \log \left (2\right )^{2} + {\left (x^{3} - 3 \, x^{2}\right )} \log \left (x\right )^{2} + {\left (x - 3\right )} e^{\left (x - \frac {6}{x}\right )} + 2 \, {\left (x^{3} \log \left (2\right ) - 3 \, x^{2} \log \left (2\right ) - x\right )} \log \left (x\right ) - 6 \, \log \left (2\right )}{x - 3} + \frac {10}{x^{2} - 6 \, x + 9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx=\frac {x^{4} \log \left (2 \, x\right )^{2} - 6 \, x^{3} \log \left (2 \, x\right )^{2} + 9 \, x^{2} \log \left (2 \, x\right )^{2} + x^{2} e^{\left (\frac {x^{2} - 6}{x}\right )} - 2 \, x^{2} \log \left (x\right ) - 6 \, x e^{\left (\frac {x^{2} - 6}{x}\right )} - 6 \, x \log \left (2 \, x\right ) + 12 \, x \log \left (x\right ) + 9 \, e^{\left (\frac {x^{2} - 6}{x}\right )} + 18 \, \log \left (2 \, x\right ) - 18 \, \log \left (x\right ) + 1}{x^{2} - 6 \, x + 9} \]
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Time = 11.71 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-20 x^2+12 x^3-2 x^4+e^{\frac {-6+x^2}{x}} \left (-162+162 x-81 x^2+33 x^3-9 x^4+x^5\right )+\left (-18 x^2-48 x^3+54 x^4-18 x^5+2 x^6\right ) \log (2 x)+\left (-54 x^3+54 x^4-18 x^5+2 x^6\right ) \log ^2(2 x)}{-27 x^2+27 x^3-9 x^4+x^5} \, dx={\mathrm {e}}^{x-\frac {6}{x}}-2\,\ln \left (x\right )+\frac {1}{x^2-6\,x+9}-\frac {6\,\ln \left (2\,x\right )}{x-3}+x^2\,{\ln \left (2\,x\right )}^2 \]
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