Integrand size = 112, antiderivative size = 30 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=5-x+x \left (-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \]
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\[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=\int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {3+3 x+3 (x+\log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )-\left (1+2 x+e^x (1+x)\right ) (x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ & = \int \left (-e^x (1+x)+\frac {3+3 x+3 x \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+3 \log (x) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )-x \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-2 x^2 \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-\log (x) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-2 x \log (x) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}\right ) \, dx \\ & = -\int e^x (1+x) \, dx+\int \frac {3+3 x+3 x \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+3 \log (x) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )-x \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-2 x^2 \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-\log (x) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )-2 x \log (x) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ & = -e^x (1+x)+\int e^x \, dx+\int \frac {\frac {3 (1+x)}{(x+\log (x)) \log (x+\log (x))}-\log \left (\frac {3}{\log (x+\log (x))}\right ) \left (-3+(1+2 x) \log \left (\frac {3}{\log (x+\log (x))}\right )\right )}{\log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ & = e^x-e^x (1+x)+\int \left (-1-2 x+\frac {3 (1+x)}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \, dx \\ & = e^x-x-x^2-e^x (1+x)+3 \int \frac {1+x}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ & = e^x-x-x^2-e^x (1+x)+3 \int \left (\frac {1}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}+\frac {x}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}\right ) \, dx+3 \int \frac {1}{\log \left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ & = e^x-x-x^2-e^x (1+x)+3 \int \frac {1}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx+3 \int \frac {x}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx+3 \int \frac {1}{\log \left (\frac {3}{\log (x+\log (x))}\right )} \, dx \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=x \left (-1-e^x-x+\frac {3}{\log \left (\frac {3}{\log (x+\log (x))}\right )}\right ) \]
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Time = 21.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
default | \(-{\mathrm e}^{x} x -x^{2}-x +\frac {3 x}{\ln \left (3\right )-\ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\) | \(31\) |
parts | \(-{\mathrm e}^{x} x -x^{2}-x +\frac {3 x}{\ln \left (3\right )-\ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\) | \(31\) |
risch | \(-x^{2}-{\mathrm e}^{x} x -x -\frac {6 i x}{-2 i \ln \left (3\right )+2 i \ln \left (\ln \left (x +\ln \left (x \right )\right )\right )}\) | \(36\) |
parallelrisch | \(\frac {-2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) x^{2}-2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) {\mathrm e}^{x} x -2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right ) x +6 x}{2 \ln \left (\frac {3}{\ln \left (x +\ln \left (x \right )\right )}\right )}\) | \(62\) |
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Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {{\left (x^{2} + x e^{x} + x\right )} \log \left (\frac {3}{\log \left (x + \log \left (x\right )\right )}\right ) - 3 \, x}{\log \left (\frac {3}{\log \left (x + \log \left (x\right )\right )}\right )} \]
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Time = 1.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=- x^{2} - x e^{x} - x + \frac {3 x}{\log {\left (\frac {3}{\log {\left (x + \log {\left (x \right )} \right )}} \right )}} \]
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Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {x^{2} \log \left (3\right ) + x e^{x} \log \left (3\right ) + x {\left (\log \left (3\right ) - 3\right )} - {\left (x^{2} + x e^{x} + x\right )} \log \left (\log \left (x + \log \left (x\right )\right )\right )}{\log \left (3\right ) - \log \left (\log \left (x + \log \left (x\right )\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (26) = 52\).
Time = 0.46 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=-\frac {x^{2} \log \left (3\right ) + x e^{x} \log \left (3\right ) - x^{2} \log \left (\log \left (x + \log \left (x\right )\right )\right ) - x e^{x} \log \left (\log \left (x + \log \left (x\right )\right )\right ) + x \log \left (3\right ) - x \log \left (\log \left (x + \log \left (x\right )\right )\right ) - 3 \, x}{\log \left (3\right ) - \log \left (\log \left (x + \log \left (x\right )\right )\right )} \]
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Time = 16.90 (sec) , antiderivative size = 204, normalized size of antiderivative = 6.80 \[ \int \frac {3+3 x+(3 x+3 \log (x)) \log (x+\log (x)) \log \left (\frac {3}{\log (x+\log (x))}\right )+\left (-x-2 x^2+e^x \left (-x-x^2\right )+\left (-1+e^x (-1-x)-2 x\right ) \log (x)\right ) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )}{(x+\log (x)) \log (x+\log (x)) \log ^2\left (\frac {3}{\log (x+\log (x))}\right )} \, dx=\frac {3\,x+\frac {3\,x\,\ln \left (x+\ln \left (x\right )\right )\,\ln \left (\frac {3}{\ln \left (x+\ln \left (x\right )\right )}\right )\,\left (x+\ln \left (x\right )\right )}{x+1}}{\ln \left (\frac {3}{\ln \left (x+\ln \left (x\right )\right )}\right )}-\ln \left (x+\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (\frac {3\,\left (x^3+2\,x^2+x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-\frac {3\,x^4+12\,x^3+12\,x^2+3\,x}{x\,{\left (x+1\right )}^2}+\frac {6\,x^4+12\,x^3+6\,x^2}{x\,{\left (x+1\right )}^2}+\frac {3\,\left (x^3+3\,x^2+2\,x\right )}{x\,{\left (x+1\right )}^2}-\frac {3\,x^2+3\,x}{x\,{\left (x+1\right )}^2}\right )-x-x\,{\mathrm {e}}^x-x^2 \]
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