Integrand size = 88, antiderivative size = 25 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \]
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\[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )\right )}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ & = \int \frac {e^{-x} \left (-\frac {4 \left (-64+e^{4+4 x}\right ) x}{e^{4+4 x}-256 x}-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ & = \int \left (\frac {256 e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx \\ & = 256 \int \frac {e^{-x} (1-4 x) x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} \left (-4 x+\log \left (\frac {1}{256} e^{4+4 x}-x\right )-x \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ & = 256 \int \left (\frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {4 e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx+\int \frac {e^{-x} \left (-4 x-(-1+x) \log \left (\frac {1}{256} e^{4+4 x}-x\right )\right )}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ & = 256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (-\frac {4 e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}+\frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx \\ & = -\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x} (1-x)}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ & = -\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \left (\frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}-\frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}\right ) \, dx \\ & = -\left (4 \int \frac {e^{-x} x}{\log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx\right )+256 \int \frac {e^{-x} x}{\left (e^{4+4 x}-256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+1024 \int \frac {e^{-x} x^2}{\left (-e^{4+4 x}+256 x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx+\int \frac {e^{-x}}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx-\int \frac {e^{-x} x}{\log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\frac {e^{-x} x}{\log \left (\frac {1}{256} e^{4+4 x}-x\right )} \]
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Time = 3.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {x \,{\mathrm e}^{-x}}{\ln \left (\frac {{\mathrm e}^{4+4 x}}{256}-x \right )}\) | \(22\) |
parallelrisch | \(\frac {x \,{\mathrm e}^{-x}}{\ln \left (\frac {{\mathrm e}^{4+4 x}}{256}-x \right )}\) | \(22\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\frac {x e^{\left (-x\right )}}{\log \left (-x + \frac {1}{256} \, e^{\left (4 \, x + 4\right )}\right )} \]
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Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\frac {x e^{- x}}{\log {\left (- x + \frac {e^{4} e^{4 x}}{256} \right )}} \]
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Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=-\frac {x}{8 \, e^{x} \log \left (2\right ) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=-\frac {x}{8 \, e^{x} \log \left (2\right ) - e^{x} \log \left (-256 \, x + e^{\left (4 \, x + 4\right )}\right )} \]
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Time = 8.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.24 \[ \int \frac {256 x-4 e^{4+4 x} x+\left (e^{4+4 x} (1-x)-256 x+256 x^2\right ) \log \left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )}{\left (e^{4+5 x}-256 e^x x\right ) \log ^2\left (\frac {1}{256} \left (e^{4+4 x}-256 x\right )\right )} \, dx=\frac {x\,{\mathrm {e}}^{-x}-\frac {{\mathrm {e}}^{-x}\,\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )\,\left (256\,x-{\mathrm {e}}^{4\,x+4}\right )\,\left (x-1\right )}{4\,\left ({\mathrm {e}}^{4\,x+4}-64\right )}}{\ln \left (\frac {{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4}{256}-x\right )}+{\mathrm {e}}^{-x}\,\left (x-x^2\right )+\frac {{\mathrm {e}}^{3\,x}\,\left (x^2-\frac {5\,x}{4}+\frac {1}{4}\right )}{{\mathrm {e}}^{4\,x}-64\,{\mathrm {e}}^{-4}} \]
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