Integrand size = 183, antiderivative size = 20 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{-\frac {1}{2}+x-\frac {x^4}{\left (-2+\log \left (x^2\right )\right )^4}} \]
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\[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=\int \frac {\exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}\right ) \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) \left (32-16 x^3-\left (80-4 x^3\right ) \log \left (x^2\right )+80 \log ^2\left (x^2\right )-40 \log ^3\left (x^2\right )+10 \log ^4\left (x^2\right )-\log ^5\left (x^2\right )\right )}{\left (2-\log \left (x^2\right )\right )^5} \, dx \\ & = \int \left (\exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right )+\frac {8 \exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) x^3}{\left (-2+\log \left (x^2\right )\right )^5}-\frac {4 \exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) x^3}{\left (-2+\log \left (x^2\right )\right )^4}\right ) \, dx \\ & = -\left (4 \int \frac {\exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) x^3}{\left (-2+\log \left (x^2\right )\right )^4} \, dx\right )+8 \int \frac {\exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) x^3}{\left (-2+\log \left (x^2\right )\right )^5} \, dx+\int \exp \left (\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{2 \left (-2+\log \left (x^2\right )\right )^4}\right ) \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(20)=40\).
Time = 0.41 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{-\frac {1}{2}+x+\frac {-32+64 x-x^4}{\left (-2+\log \left (x^2\right )\right )^4}+\frac {16 (-1+2 x)}{\left (-2+\log \left (x^2\right )\right )^3}} \left (x^2\right )^{-\frac {16 (-1+2 x)}{\left (-2+\log \left (x^2\right )\right )^4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(17)=34\).
Time = 2.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 4.30
method | result | size |
risch | \({\mathrm e}^{\frac {2 x \ln \left (x^{2}\right )^{4}-\ln \left (x^{2}\right )^{4}-16 x \ln \left (x^{2}\right )^{3}-2 x^{4}+8 \ln \left (x^{2}\right )^{3}+48 x \ln \left (x^{2}\right )^{2}-24 \ln \left (x^{2}\right )^{2}-64 x \ln \left (x^{2}\right )+32 \ln \left (x^{2}\right )+32 x -16}{2 {\left (-2+\ln \left (x^{2}\right )\right )}^{4}}}\) | \(86\) |
parallelrisch | \({\mathrm e}^{\frac {\left (-1+2 x \right ) \ln \left (x^{2}\right )^{4}+\left (-16 x +8\right ) \ln \left (x^{2}\right )^{3}+\left (48 x -24\right ) \ln \left (x^{2}\right )^{2}+\left (-64 x +32\right ) \ln \left (x^{2}\right )-2 x^{4}+32 x -16}{2 \ln \left (x^{2}\right )^{4}-16 \ln \left (x^{2}\right )^{3}+48 \ln \left (x^{2}\right )^{2}-64 \ln \left (x^{2}\right )+32}}\) | \(92\) |
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.70 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {{\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{4} - 2 \, x^{4} - 8 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{3} + 24 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right )^{2} - 32 \, {\left (2 \, x - 1\right )} \log \left (x^{2}\right ) + 32 \, x - 16}{2 \, {\left (\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).
Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.50 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\frac {- 2 x^{4} + 32 x + \left (8 - 16 x\right ) \log {\left (x^{2} \right )}^{3} + \left (32 - 64 x\right ) \log {\left (x^{2} \right )} + \left (2 x - 1\right ) \log {\left (x^{2} \right )}^{4} + \left (48 x - 24\right ) \log {\left (x^{2} \right )}^{2} - 16}{2 \log {\left (x^{2} \right )}^{4} - 16 \log {\left (x^{2} \right )}^{3} + 48 \log {\left (x^{2} \right )}^{2} - 64 \log {\left (x^{2} \right )} + 32}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 322 vs. \(2 (17) = 34\).
Time = 0.53 (sec) , antiderivative size = 322, normalized size of antiderivative = 16.10 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {x \log \left (x\right )^{4}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {x^{4}}{16 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}} - \frac {4 \, x \log \left (x\right )^{3}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {\log \left (x\right )^{4}}{2 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}} + \frac {6 \, x \log \left (x\right )^{2}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {2 \, \log \left (x\right )^{3}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {4 \, x \log \left (x\right )}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {3 \, \log \left (x\right )^{2}}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {x}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} + \frac {2 \, \log \left (x\right )}{\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1} - \frac {1}{2 \, {\left (\log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (17) = 34\).
Time = 2.13 (sec) , antiderivative size = 427, normalized size of antiderivative = 21.35 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx=e^{\left (\frac {x \log \left (x^{2}\right )^{4}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {x^{4}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {8 \, x \log \left (x^{2}\right )^{3}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {\log \left (x^{2}\right )^{4}}{2 \, {\left (\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16\right )}} + \frac {24 \, x \log \left (x^{2}\right )^{2}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {4 \, \log \left (x^{2}\right )^{3}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {32 \, x \log \left (x^{2}\right )}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {12 \, \log \left (x^{2}\right )^{2}}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {16 \, x}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} + \frac {16 \, \log \left (x^{2}\right )}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16} - \frac {8}{\log \left (x^{2}\right )^{4} - 8 \, \log \left (x^{2}\right )^{3} + 24 \, \log \left (x^{2}\right )^{2} - 32 \, \log \left (x^{2}\right ) + 16}\right )} \]
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Time = 9.52 (sec) , antiderivative size = 419, normalized size of antiderivative = 20.95 \[ \int \frac {e^{\frac {-16+32 x-2 x^4+(32-64 x) \log \left (x^2\right )+(-24+48 x) \log ^2\left (x^2\right )+(8-16 x) \log ^3\left (x^2\right )+(-1+2 x) \log ^4\left (x^2\right )}{32-64 \log \left (x^2\right )+48 \log ^2\left (x^2\right )-16 \log ^3\left (x^2\right )+2 \log ^4\left (x^2\right )}} \left (-32+16 x^3+\left (80-4 x^3\right ) \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )\right )}{-32+80 \log \left (x^2\right )-80 \log ^2\left (x^2\right )+40 \log ^3\left (x^2\right )-10 \log ^4\left (x^2\right )+\log ^5\left (x^2\right )} \, dx={\mathrm {e}}^{-\frac {2\,x^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {16}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {2\,x\,{\ln \left (x^2\right )}^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {16\,x\,{\ln \left (x^2\right )}^3}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {48\,x\,{\ln \left (x^2\right )}^2}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {32\,x}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {{\ln \left (x^2\right )}^4}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{\frac {8\,{\ln \left (x^2\right )}^3}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\mathrm {e}}^{-\frac {24\,{\ln \left (x^2\right )}^2}{2\,{\ln \left (x^2\right )}^4-16\,{\ln \left (x^2\right )}^3+48\,{\ln \left (x^2\right )}^2-64\,\ln \left (x^2\right )+32}}\,{\left (\frac {1}{x^{32}}\right )}^{\frac {2\,x-1}{{\ln \left (x^2\right )}^4-8\,{\ln \left (x^2\right )}^3+24\,{\ln \left (x^2\right )}^2-32\,\ln \left (x^2\right )+16}} \]
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