Integrand size = 39, antiderivative size = 20 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=e^{e^x+e^{\frac {1}{4}+2 x}}+x \log (x) \]
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Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2320, 2268, 2332} \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=e^{e^x+e^{2 x+\frac {1}{4}}}+x \log (x) \]
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Rule 2268
Rule 2320
Rule 2332
Rubi steps \begin{align*} \text {integral}& = x+\int e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right ) \, dx+\int \log (x) \, dx \\ & = x \log (x)+\text {Subst}\left (\int e^{x+\sqrt [4]{e} x^2} \left (1+2 \sqrt [4]{e} x\right ) \, dx,x,e^x\right ) \\ & = e^{e^x+e^{\frac {1}{4}+2 x}}+x \log (x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=e^{e^x+e^{\frac {1}{4}+2 x}}+x \log (x) \]
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Time = 0.11 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80
method | result | size |
default | \(x \ln \left (x \right )+{\mathrm e}^{{\mathrm e}^{2 x +\frac {1}{4}}+{\mathrm e}^{x}}\) | \(16\) |
risch | \(x \ln \left (x \right )+{\mathrm e}^{{\mathrm e}^{2 x +\frac {1}{4}}+{\mathrm e}^{x}}\) | \(16\) |
parallelrisch | \(x \ln \left (x \right )+{\mathrm e}^{{\mathrm e}^{2 x +\frac {1}{4}}+{\mathrm e}^{x}}\) | \(16\) |
norman | \(x \ln \left (x \right )+{\mathrm e}^{{\mathrm e}^{\frac {1}{4}} {\mathrm e}^{2 x}+{\mathrm e}^{x}}\) | \(17\) |
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Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=x \log \left (x\right ) + e^{\left (e^{\left (2 \, x + \frac {1}{4}\right )} + e^{x}\right )} \]
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Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=x \log {\left (x \right )} + e^{e^{\frac {1}{4}} e^{2 x} + e^{x}} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=x \log \left (x\right ) + e^{\left (e^{\left (2 \, x + \frac {1}{4}\right )} + e^{x}\right )} \]
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx=x \log \left (x\right ) + e^{\left (e^{\left (2 \, x + \frac {1}{4}\right )} + e^{x}\right )} \]
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Time = 8.85 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \left (1+e^{e^x+e^{\frac {1}{4} (1+8 x)}} \left (e^x+2 e^{\frac {1}{4} (1+8 x)}\right )+\log (x)\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{1/4}}+x\,\ln \left (x\right ) \]
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