Integrand size = 95, antiderivative size = 21 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \log ^2\left (1-e^{-x}+x+x^6\right )}{x} \]
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\[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {16 \left (\frac {2 x \left (1+e^x \left (1+6 x^5\right )\right )}{-1+e^x \left (1+x+x^6\right )}-\log \left (1-e^{-x}+x+x^6\right )\right ) \log \left (1-e^{-x}+x+x^6\right )}{x^2} \, dx \\ & = 16 \int \frac {\left (\frac {2 x \left (1+e^x \left (1+6 x^5\right )\right )}{-1+e^x \left (1+x+x^6\right )}-\log \left (1-e^{-x}+x+x^6\right )\right ) \log \left (1-e^{-x}+x+x^6\right )}{x^2} \, dx \\ & = 16 \int \left (\frac {2 \left (2+x+6 x^5+x^6\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}-\frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-2 x-12 x^6+\log \left (1-e^{-x}+x+x^6\right )+x \log \left (1-e^{-x}+x+x^6\right )+x^6 \log \left (1-e^{-x}+x+x^6\right )\right )}{x^2 \left (1+x+x^6\right )}\right ) \, dx \\ & = -\left (16 \int \frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-2 x-12 x^6+\log \left (1-e^{-x}+x+x^6\right )+x \log \left (1-e^{-x}+x+x^6\right )+x^6 \log \left (1-e^{-x}+x+x^6\right )\right )}{x^2 \left (1+x+x^6\right )} \, dx\right )+32 \int \frac {\left (2+x+6 x^5+x^6\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = -\left (16 \int \frac {\log \left (1-e^{-x}+x+x^6\right ) \left (-\frac {2 \left (x+6 x^6\right )}{1+x+x^6}+\log \left (1-e^{-x}+x+x^6\right )\right )}{x^2} \, dx\right )-32 \int \frac {\left (1+e^x \left (1+6 x^5\right )\right ) \left (-2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1-e^x \left (1+x+x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = -\left (16 \int \left (-\frac {2 \left (1+6 x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right )}+\frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2}\right ) \, dx\right )-32 \int \left (\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6}+\frac {\left (2+x+6 x^5+x^6\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}\right ) \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = -\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \frac {\left (1+6 x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{x \left (1+x+x^6\right )} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6} \, dx-32 \int \frac {\left (2+x+6 x^5+x^6\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = -\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \left (\frac {\log \left (1-e^{-x}+x+x^6\right )}{x}-\frac {\left (1-6 x^4+x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{1+x+x^6}\right ) \, dx-32 \int \left (\frac {2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{-1+e^x+e^x x+e^x x^6}+\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )}\right ) \, dx-32 \int \left (\frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6}-\frac {\left (1+6 x^5\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{1+x+x^6}\right ) \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = -\left (16 \int \frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x^2} \, dx\right )+32 \int \frac {\log \left (1-e^{-x}+x+x^6\right )}{x} \, dx-32 \int \frac {\left (1-6 x^4+x^5\right ) \log \left (1-e^{-x}+x+x^6\right )}{1+x+x^6} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{1+x+x^6} \, dx-32 \int \frac {2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{-1+e^x+e^x x+e^x x^6} \, dx-32 \int \frac {\left (1+6 x^5\right ) \left (2 \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+6 \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx-\int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx\right )}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+32 \int \frac {\left (1+6 x^5\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx}{1+x+x^6} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx-\left (32 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^5}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx+\left (64 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {1}{x \left (-1+e^x \left (1+x+x^6\right )\right )} \, dx+\left (192 \log \left (1-e^{-x}+x+x^6\right )\right ) \int \frac {x^4}{\left (1+x+x^6\right ) \left (-1+e^x+e^x x+e^x x^6\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(21)=42\).
Time = 0.73 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.76 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=16 \left (2 x+2 \log \left (1-e^{-x}+x+x^6\right )+\frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x}-2 \log \left (1-e^x \left (1+x+x^6\right )\right )\right ) \]
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Time = 1.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {16 \ln \left (-{\mathrm e}^{-x}+x^{6}+x +1\right )^{2}}{x}\) | \(21\) |
parallelrisch | \(\frac {16 \ln \left (-{\mathrm e}^{-x}+x^{6}+x +1\right )^{2}}{x}\) | \(21\) |
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \]
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Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \log {\left (x^{6} + x + 1 - e^{- x} \right )}^{2}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, {\left (x^{2} + \log \left ({\left (x^{6} + x + 1\right )} e^{x} - 1\right )^{2}\right )}}{x} - 32 \, \log \left (x^{6} + x + 1\right ) - 32 \, \log \left (\frac {{\left (x^{6} + x + 1\right )} e^{x} - 1}{x^{6} + x + 1}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \]
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Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16\,{\ln \left (x-{\mathrm {e}}^{-x}+x^6+1\right )}^2}{x} \]
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