Integrand size = 209, antiderivative size = 27 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} e^{\frac {1}{-\left (3-e^{3 x}-x\right )^2+\log (x)}} \]
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Time = 3.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6820, 12, 6838} \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} e^{-\frac {1}{\left (x+e^{3 x}-3\right )^2-\log (x)}} \]
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Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^{-\frac {1}{\left (-3+e^{3 x}+x\right )^2-\log (x)}} \left (-1+2 \left (-3-8 e^{3 x}+3 e^{6 x}\right ) x+\left (2+6 e^{3 x}\right ) x^2\right )}{4 x \left (\left (-3+e^{3 x}+x\right )^2-\log (x)\right )^2} \, dx \\ & = \frac {3}{4} \int \frac {e^{-\frac {1}{\left (-3+e^{3 x}+x\right )^2-\log (x)}} \left (-1+2 \left (-3-8 e^{3 x}+3 e^{6 x}\right ) x+\left (2+6 e^{3 x}\right ) x^2\right )}{x \left (\left (-3+e^{3 x}+x\right )^2-\log (x)\right )^2} \, dx \\ & = \frac {3}{4} e^{-\frac {1}{\left (-3+e^{3 x}+x\right )^2-\log (x)}} \\ \end{align*}
Time = 4.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} e^{-\frac {1}{\left (-3+e^{3 x}+x\right )^2-\log (x)}} \]
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Time = 41.55 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {3 \,{\mathrm e}^{\frac {1}{-2 x \,{\mathrm e}^{3 x}-x^{2}-{\mathrm e}^{6 x}+\ln \left (x \right )+6 \,{\mathrm e}^{3 x}+6 x -9}}}{4}\) | \(37\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{\frac {1}{-2 x \,{\mathrm e}^{3 x}-x^{2}-{\mathrm e}^{6 x}+\ln \left (x \right )+6 \,{\mathrm e}^{3 x}+6 x -9}}}{4}\) | \(39\) |
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} \, e^{\left (-\frac {1}{x^{2} + 2 \, {\left (x - 3\right )} e^{\left (3 \, x\right )} - 6 \, x + e^{\left (6 \, x\right )} - \log \left (x\right ) + 9}\right )} \]
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Time = 4.62 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3 e^{\frac {1}{- x^{2} + 6 x + \left (6 - 2 x\right ) e^{3 x} - e^{6 x} + \log {\left (x \right )} - 9}}}{4} \]
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Time = 1.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} \, e^{\left (-\frac {1}{x^{2} + 2 \, {\left (x - 3\right )} e^{\left (3 \, x\right )} - 6 \, x + e^{\left (6 \, x\right )} - \log \left (x\right ) + 9}\right )} \]
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Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3}{4} \, e^{\left (-\frac {1}{x^{2} + 2 \, x e^{\left (3 \, x\right )} - 6 \, x + e^{\left (6 \, x\right )} - 6 \, e^{\left (3 \, x\right )} - \log \left (x\right ) + 9}\right )} \]
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Time = 14.72 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\frac {1}{-9-e^{6 x}+e^{3 x} (6-2 x)+6 x-x^2+\log (x)}} \left (-3-18 x+18 e^{6 x} x+6 x^2+e^{3 x} \left (-48 x+18 x^2\right )\right )}{324 x+4 e^{12 x} x-432 x^2+216 x^3-48 x^4+4 x^5+e^{9 x} \left (-48 x+16 x^2\right )+e^{6 x} \left (216 x-144 x^2+24 x^3\right )+e^{3 x} \left (-432 x+432 x^2-144 x^3+16 x^4\right )+\left (-72 x-8 e^{6 x} x+48 x^2-8 x^3+e^{3 x} \left (48 x-16 x^2\right )\right ) \log (x)+4 x \log ^2(x)} \, dx=\frac {3\,{\mathrm {e}}^{-\frac {1}{{\mathrm {e}}^{6\,x}-6\,{\mathrm {e}}^{3\,x}-6\,x-\ln \left (x\right )+2\,x\,{\mathrm {e}}^{3\,x}+x^2+9}}}{4} \]
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