Integrand size = 126, antiderivative size = 35 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=e^{e^2-\left (x-\log \left (\frac {1}{x}-x\right )\right )^2}-4 (4-x)^2 x^4 \]
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\[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=\int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{x \left (-1+x^2\right )} \, dx \\ & = \int \left (\frac {256 x^3}{-1+x^2}-\frac {160 x^4}{-1+x^2}-\frac {232 x^5}{-1+x^2}+\frac {160 x^6}{-1+x^2}-\frac {24 x^7}{-1+x^2}+\frac {2 e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \left (-x+\log \left (\frac {1}{x}-x\right )\right )}{x \left (1-x^2\right )}\right ) \, dx \\ & = 2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \left (-x+\log \left (\frac {1}{x}-x\right )\right )}{x \left (1-x^2\right )} \, dx-24 \int \frac {x^7}{-1+x^2} \, dx-160 \int \frac {x^4}{-1+x^2} \, dx+160 \int \frac {x^6}{-1+x^2} \, dx-232 \int \frac {x^5}{-1+x^2} \, dx+256 \int \frac {x^3}{-1+x^2} \, dx \\ & = 2 \int \left (\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right )}{-1+x^2}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \log \left (\frac {1-x^2}{x}\right )}{x \left (1-x^2\right )}\right ) \, dx-12 \text {Subst}\left (\int \frac {x^3}{-1+x} \, dx,x,x^2\right )-116 \text {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,x^2\right )+128 \text {Subst}\left (\int \frac {x}{-1+x} \, dx,x,x^2\right )-160 \int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx+160 \int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx \\ & = 32 x^5+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right )}{-1+x^2} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \left (1+x+x^2-x^3\right ) \log \left (\frac {1-x^2}{x}\right )}{x \left (1-x^2\right )} \, dx-12 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x+x^2\right ) \, dx,x,x^2\right )-116 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,x^2\right )+128 \text {Subst}\left (\int \left (1+\frac {1}{-1+x}\right ) \, dx,x,x^2\right ) \\ & = -64 x^4+32 x^5-4 x^6+2 \int \left (e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}-e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x}+\frac {2 e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{-1+x^2}\right ) \, dx+2 \int \left (e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x}+\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x}\right ) \, dx \\ & = -64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx+4 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{-1+x^2} \, dx \\ & = -64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx+4 \int \left (-\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{2 (1-x)}-\frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{2 (1+x)}\right ) \, dx \\ & = -64 x^4+32 x^5-4 x^6+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{1-x} \, dx-2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} x \left (\frac {1-x^2}{x}\right )^{2 x} \, dx-2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x}}{1+x} \, dx+2 \int e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right ) \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{-1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{1-x} \, dx+2 \int \frac {e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1-x^2}{x}\right )^{2 x} \log \left (\frac {1-x^2}{x}\right )}{x} \, dx \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=e^{e^2-x^2-\log ^2\left (\frac {1}{x}-x\right )} \left (\frac {1}{x}-x\right )^{2 x}-4 (-4+x)^2 x^4 \]
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Time = 1.89 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57
method | result | size |
parallelrisch | \(-4 x^{6}+32 x^{5}-64 x^{4}+{\mathrm e}^{-\ln \left (-\frac {x^{2}-1}{x}\right )^{2}+2 x \ln \left (-\frac {x^{2}-1}{x}\right )+{\mathrm e}^{2}-x^{2}}\) | \(55\) |
default | \({\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+2 x \ln \left (\frac {-x^{2}+1}{x}\right )+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(57\) |
parts | \({\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+2 x \ln \left (\frac {-x^{2}+1}{x}\right )+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(57\) |
risch | \(\left (\frac {-x^{2}+1}{x}\right )^{2 x} {\mathrm e}^{-\ln \left (\frac {-x^{2}+1}{x}\right )^{2}+{\mathrm e}^{2}-x^{2}}-4 x^{6}+32 x^{5}-64 x^{4}\) | \(58\) |
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Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-\frac {x^{2} - 1}{x}\right ) - \log \left (-\frac {x^{2} - 1}{x}\right )^{2} + e^{2}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=- 4 x^{6} + 32 x^{5} - 64 x^{4} + e^{- x^{2} + 2 x \log {\left (\frac {1 - x^{2}}{x} \right )} - \log {\left (\frac {1 - x^{2}}{x} \right )}^{2} + e^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (31) = 62\).
Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.86 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} - 2 \, x \log \left (x\right ) + 2 \, \log \left (x + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, x \log \left (-x + 1\right ) - 2 \, \log \left (x + 1\right ) \log \left (-x + 1\right ) + 2 \, \log \left (x\right ) \log \left (-x + 1\right ) - \log \left (-x + 1\right )^{2} + e^{2}\right )} \]
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Time = 0.76 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=-4 \, x^{6} + 32 \, x^{5} - 64 \, x^{4} + e^{\left (-x^{2} + 2 \, x \log \left (-x + \frac {1}{x}\right ) - \log \left (-x + \frac {1}{x}\right )^{2} + e^{2}\right )} \]
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Time = 14.50 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57 \[ \int \frac {256 x^4-160 x^5-232 x^6+160 x^7-24 x^8+e^{e^2-x^2+2 x \log \left (\frac {1-x^2}{x}\right )-\log ^2\left (\frac {1-x^2}{x}\right )} \left (2 x+2 x^2+2 x^3-2 x^4+\left (-2-2 x-2 x^2+2 x^3\right ) \log \left (\frac {1-x^2}{x}\right )\right )}{-x+x^3} \, dx=32\,x^5-64\,x^4-4\,x^6+{\mathrm {e}}^{-x^2-{\ln \left (-\frac {x^2-1}{x}\right )}^2+{\mathrm {e}}^2}\,{\left (-\frac {x^2-1}{x}\right )}^{2\,x} \]
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