Integrand size = 191, antiderivative size = 31 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x (-x+\log (x))^2+\log \left (-x-\frac {1}{625} \log ^2\left (e^x-x\right )\right ) \]
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\[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=\int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (-1+e^x\right ) \log \left (e^x-x\right )-\left (e^x-x\right ) \log ^2\left (e^x-x\right ) \left (x (-2+3 x)+(2-4 x) \log (x)+\log ^2(x)\right )-625 \left (e^x-x\right ) \left (-1-2 x^2+3 x^3+2 (1-2 x) x \log (x)+x \log ^2(x)\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx \\ & = \int \left (\frac {2 (-1+x) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}+\frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )-1250 x \log (x)+2500 x^2 \log (x)-2 \log ^2\left (e^x-x\right ) \log (x)+4 x \log ^2\left (e^x-x\right ) \log (x)-625 x \log ^2(x)-\log ^2\left (e^x-x\right ) \log ^2(x)}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx \\ & = 2 \int \frac {(-1+x) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+\int \frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )-1250 x \log (x)+2500 x^2 \log (x)-2 \log ^2\left (e^x-x\right ) \log (x)+4 x \log ^2\left (e^x-x\right ) \log (x)-625 x \log ^2(x)-\log ^2\left (e^x-x\right ) \log ^2(x)}{625 x+\log ^2\left (e^x-x\right )} \, dx \\ & = 2 \int \left (-\frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}+\frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}\right ) \, dx+\int \frac {2 \log \left (e^x-x\right )-\log ^2\left (e^x-x\right ) \left (x (-2+3 x)+(2-4 x) \log (x)+\log ^2(x)\right )-625 \left (-1-2 x^2+3 x^3+2 (1-2 x) x \log (x)+x \log ^2(x)\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx \\ & = -\left (2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx\right )+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+\int \left (\frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}+2 (-1+2 x) \log (x)-\log ^2(x)\right ) \, dx \\ & = -\left (2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx\right )+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int (-1+2 x) \log (x) \, dx+\int \frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx-\int \log ^2(x) \, dx \\ & = -2 x \log (x)+2 x^2 \log (x)-x \log ^2(x)-2 \int (-1+x) \, dx-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \log (x) \, dx+\int \left (-x (-2+3 x)+\frac {625+2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx \\ & = -x^2+2 x^2 \log (x)-x \log ^2(x)-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx-\int x (-2+3 x) \, dx+\int \frac {625+2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx \\ & = -x^2+2 x^2 \log (x)-x \log ^2(x)-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx-\int \left (-2 x+3 x^2\right ) \, dx+\int \left (\frac {625}{625 x+\log ^2\left (e^x-x\right )}+\frac {2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx \\ & = -x^3+2 x^2 \log (x)-x \log ^2(x)+2 \int \frac {\log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+625 \int \frac {1}{625 x+\log ^2\left (e^x-x\right )} \, dx \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^3+2 x^2 \log (x)-x \log ^2(x)+\log \left (625 x+\log ^2\left (e^x-x\right )\right ) \]
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Time = 1.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-x^{3}+2 x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+\ln \left (\ln \left ({\mathrm e}^{x}-x \right )^{2}+625 x \right )\) | \(35\) |
parallelrisch | \(-x^{3}+2 x^{2} \ln \left (x \right )-x \ln \left (x \right )^{2}+\ln \left (\frac {\ln \left ({\mathrm e}^{x}-x \right )^{2}}{625}+x \right )\) | \(35\) |
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Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=- x^{3} + 2 x^{2} \log {\left (x \right )} - x \log {\left (x \right )}^{2} + \log {\left (625 x + \log {\left (- x + e^{x} \right )}^{2} \right )} \]
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Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \]
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Time = 0.34 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=-x^{3} + 2 \, x^{2} \log \left (x\right ) - x \log \left (x\right )^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \]
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Time = 9.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx=\ln \left ({\ln \left ({\mathrm {e}}^x-x\right )}^2+625\,x\right )-x\,{\ln \left (x\right )}^2+2\,x^2\,\ln \left (x\right )-x^3 \]
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