Integrand size = 78, antiderivative size = 20 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \]
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\[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {7 \left (e^3-e^{x^2}-e^{x^2} \left (-1+2 x^2\right ) \log (x)\right )}{5 \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx \\ & = \frac {7}{5} \int \frac {e^3-e^{x^2}-e^{x^2} \left (-1+2 x^2\right ) \log (x)}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx \\ & = \frac {7}{5} \int \left (-\frac {1-\log (x)+2 x^2 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )}+\frac {e^3+x-x \log (x)+2 e^3 x^2 \log (x)+2 x^3 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}\right ) \, dx \\ & = -\left (\frac {7}{5} \int \frac {1-\log (x)+2 x^2 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )} \, dx\right )+\frac {7}{5} \int \frac {e^3+x-x \log (x)+2 e^3 x^2 \log (x)+2 x^3 \log (x)}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx \\ & = \frac {7}{5} \int \left (-\frac {x}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {2 e^3 x^2}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {2 x^3}{\left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {e^3}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}+\frac {x}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2}\right ) \, dx-\frac {7}{5} \int \left (-\frac {1}{e^3+x+e^{x^2} \log (x)}+\frac {2 x^2}{e^3+x+e^{x^2} \log (x)}+\frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )}\right ) \, dx \\ & = -\left (\frac {7}{5} \int \frac {x}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx\right )+\frac {7}{5} \int \frac {x}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx+\frac {7}{5} \int \frac {1}{e^3+x+e^{x^2} \log (x)} \, dx-\frac {7}{5} \int \frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )} \, dx+\frac {14}{5} \int \frac {x^3}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx-\frac {14}{5} \int \frac {x^2}{e^3+x+e^{x^2} \log (x)} \, dx+\frac {1}{5} \left (7 e^3\right ) \int \frac {1}{\log (x) \left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx+\frac {1}{5} \left (14 e^3\right ) \int \frac {x^2}{\left (e^3+x+e^{x^2} \log (x)\right )^2} \, dx \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 \left (e^3+x+e^{x^2} \log (x)\right )} \]
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Time = 1.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {7 x}{5 \left ({\mathrm e}^{3}+{\mathrm e}^{x^{2}} \ln \left (x \right )+x \right )}\) | \(17\) |
parallelrisch | \(\frac {7 x}{5 \left ({\mathrm e}^{3}+{\mathrm e}^{x^{2}} \ln \left (x \right )+x \right )}\) | \(17\) |
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \]
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Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 x}{5 x + 5 e^{x^{2}} \log {\left (x \right )} + 5 e^{3}} \]
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Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7 \, x}{5 \, {\left (e^{\left (x^{2}\right )} \log \left (x\right ) + x + e^{3}\right )}} \]
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Time = 9.87 (sec) , antiderivative size = 110, normalized size of antiderivative = 5.50 \[ \int \frac {7 e^3-7 e^{x^2}+e^{x^2} \left (7-14 x^2\right ) \log (x)}{5 e^6+10 e^3 x+5 x^2+e^{x^2} \left (10 e^3+10 x\right ) \log (x)+5 e^{2 x^2} \log ^2(x)} \, dx=\frac {7\,\left (2\,x^5\,{\ln \left (x\right )}^2+2\,{\mathrm {e}}^3\,x^4\,{\ln \left (x\right )}^2-x^3\,{\ln \left (x\right )}^2+x^3\,\ln \left (x\right )+{\mathrm {e}}^3\,x^2\,\ln \left (x\right )\right )}{5\,\left ({\mathrm {e}}^{x^2}+\frac {x+{\mathrm {e}}^3}{\ln \left (x\right )}\right )\,\left (2\,x^4\,{\ln \left (x\right )}^3+2\,{\mathrm {e}}^3\,x^3\,{\ln \left (x\right )}^3-x^2\,{\ln \left (x\right )}^3+x^2\,{\ln \left (x\right )}^2+{\mathrm {e}}^3\,x\,{\ln \left (x\right )}^2\right )} \]
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