Integrand size = 91, antiderivative size = 27 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=e^{x+x \left ((1+x)^4+\frac {1}{3} \left (x-\frac {x}{\log (x)}\right )\right )}+x \]
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Time = 0.90 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6874, 6820, 6838} \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=\exp \left (\frac {1}{3} x \left (3 x^4+12 x^3+18 x^2+13 x-\frac {x}{\log (x)}+6\right )\right )+x \]
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Rule 12
Rule 6820
Rule 6838
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {3 \log ^2(x)+\exp \left (\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}\right ) \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = \frac {1}{3} \int \left (3+\frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+6 \log ^2(x)+26 x \log ^2(x)+54 x^2 \log ^2(x)+48 x^3 \log ^2(x)+15 x^4 \log ^2(x)\right )}{\log ^2(x)}\right ) \, dx \\ & = x+\frac {1}{3} \int \frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+6 \log ^2(x)+26 x \log ^2(x)+54 x^2 \log ^2(x)+48 x^3 \log ^2(x)+15 x^4 \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = x+\frac {1}{3} \int \frac {\exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right ) \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx \\ & = \exp \left (\frac {1}{3} x \left (6+13 x+18 x^2+12 x^3+3 x^4-\frac {x}{\log (x)}\right )\right )+x \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=e^{2 x+\frac {13 x^2}{3}+6 x^3+4 x^4+x^5-\frac {x^2}{3 \log (x)}}+x \]
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Time = 0.22 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59
method | result | size |
parallelrisch | \(x +{\mathrm e}^{\frac {\left (3 x^{5}+12 x^{4}+18 x^{3}+13 x^{2}+6 x \right ) \ln \left (x \right )-x^{2}}{3 \ln \left (x \right )}}\) | \(43\) |
risch | \(x +{\mathrm e}^{\frac {x \left (3 x^{4} \ln \left (x \right )+12 x^{3} \ln \left (x \right )+18 x^{2} \ln \left (x \right )+13 x \ln \left (x \right )+6 \ln \left (x \right )-x \right )}{3 \ln \left (x \right )}}\) | \(45\) |
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Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=x + e^{\left (-\frac {x^{2} - {\left (3 \, x^{5} + 12 \, x^{4} + 18 \, x^{3} + 13 \, x^{2} + 6 \, x\right )} \log \left (x\right )}{3 \, \log \left (x\right )}\right )} \]
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Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=x + e^{\frac {- \frac {x^{2}}{3} + \frac {\left (3 x^{5} + 12 x^{4} + 18 x^{3} + 13 x^{2} + 6 x\right ) \log {\left (x \right )}}{3}}{\log {\left (x \right )}}} \]
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Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=x + e^{\left (x^{5} + 4 \, x^{4} + 6 \, x^{3} + \frac {13}{3} \, x^{2} + 2 \, x - \frac {x^{2}}{3 \, \log \left (x\right )}\right )} \]
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Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=x + e^{\left (\frac {3 \, x^{5} \log \left (x\right ) + 12 \, x^{4} \log \left (x\right ) + 18 \, x^{3} \log \left (x\right ) + 13 \, x^{2} \log \left (x\right ) - x^{2} + 6 \, x \log \left (x\right )}{3 \, \log \left (x\right )}\right )} \]
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Time = 13.83 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3 \log ^2(x)+e^{\frac {-x^2+\left (6 x+13 x^2+18 x^3+12 x^4+3 x^5\right ) \log (x)}{3 \log (x)}} \left (x-2 x \log (x)+\left (6+26 x+54 x^2+48 x^3+15 x^4\right ) \log ^2(x)\right )}{3 \log ^2(x)} \, dx=x+{\left ({\mathrm {e}}^{x^2}\right )}^{13/3}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{x^5}\,{\mathrm {e}}^{4\,x^4}\,{\mathrm {e}}^{6\,x^3}\,{\mathrm {e}}^{-\frac {x^2}{3\,\ln \left (x\right )}} \]
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