Integrand size = 164, antiderivative size = 31 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=-8+x+\frac {x}{1-\frac {\log ^2\left (\log \left (3 e^3\right )\right )}{(2 x-x \log (3))^2}} \]
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Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6, 2019, 28, 1828, 21, 8} \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))}+2 x \]
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Rule 6
Rule 8
Rule 21
Rule 28
Rule 1828
Rule 2019
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 (32-64 \log (3))+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+x^4 \left (32-64 \log (3)+48 \log ^2(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)\right )+x^4 \left (-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 (16-32 \log (3))+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{-8 x^4 \log ^3(3)+x^4 \log ^4(3)+x^4 \left (16-32 \log (3)+24 \log ^2(3)\right )+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \left (16-32 \log (3)+24 \log ^2(3)\right )+x^4 \left (-8 \log ^3(3)+\log ^4(3)\right )+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \left (16-32 \log (3)+24 \log ^2(3)-8 \log ^3(3)+\log ^4(3)\right )+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx \\ & = \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 (2-\log (3))^4-2 x^2 (2-\log (3))^2 \log ^2(3+\log (3))+\log ^4(3+\log (3))} \, dx \\ & = (2-\log (3))^4 \int \frac {x^4 \left (32-64 \log (3)+48 \log ^2(3)-16 \log ^3(3)+2 \log ^4(3)\right )+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{\left (x^2 (2-\log (3))^4-(2-\log (3))^2 \log ^2(3+\log (3))\right )^2} \, dx \\ & = \frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))}+\frac {(2-\log (3))^2 \int \frac {4 x^2 (2-\log (3))^2 \log ^2(3+\log (3))-4 \log ^4(3+\log (3))}{x^2 (2-\log (3))^4-(2-\log (3))^2 \log ^2(3+\log (3))} \, dx}{2 \log ^2(3+\log (3))} \\ & = \frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))}+2 \int 1 \, dx \\ & = 2 x+\frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=x \left (2+\frac {\log ^2(3+\log (3))}{x^2 (-2+\log (3))^2-\log ^2(3+\log (3))}\right ) \]
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Time = 1.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48
method | result | size |
risch | \(2 x +\frac {\ln \left (3+\ln \left (3\right )\right )^{2} x}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-\ln \left (3+\ln \left (3\right )\right )^{2}+4 x^{2}}\) | \(46\) |
norman | \(\frac {\left (2 \ln \left (3\right )^{2}-8 \ln \left (3\right )+8\right ) x^{3}-\ln \left (3+\ln \left (3\right )\right )^{2} x}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-\ln \left (3+\ln \left (3\right )\right )^{2}+4 x^{2}}\) | \(61\) |
gosper | \(\frac {x \left (2 x^{2} \ln \left (3\right )^{2}-8 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+8 x^{2}\right )}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+4 x^{2}}\) | \(68\) |
default | \(2 x +\frac {{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}}{2 \left (2-\ln \left (3\right )\right ) \left (-x \ln \left (3\right )+\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )+2 x \right )}+\frac {{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}}{2 \left (2-\ln \left (3\right )\right ) \left (-x \ln \left (3\right )-\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )+2 x \right )}\) | \(77\) |
parallelrisch | \(\frac {2 x^{3} \ln \left (3\right )^{4}-16 x^{3} \ln \left (3\right )^{3}-\ln \left (3\right )^{2} {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x +48 x^{3} \ln \left (3\right )^{2}+4 \ln \left (3\right ) {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x -64 x^{3} \ln \left (3\right )-4 {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x +32 x^{3}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+4 x^{2}\right )}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {2 \, x^{3} \log \left (3\right )^{2} - 8 \, x^{3} \log \left (3\right ) + 8 \, x^{3} - x \log \left (\log \left (3\right ) + 3\right )^{2}}{x^{2} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right ) + 4 \, x^{2} - \log \left (\log \left (3\right ) + 3\right )^{2}} \]
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Time = 1.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=2 x + \frac {x \log {\left (\log {\left (3 \right )} + 3 \right )}^{2}}{x^{2} \left (- 4 \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 4\right ) - \log {\left (\log {\left (3 \right )} + 3 \right )}^{2}} \]
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Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x \log \left (\log \left (3 \, e^{3}\right )\right )^{2}}{{\left (\log \left (3\right )^{2} - 4 \, \log \left (3\right ) + 4\right )} x^{2} - \log \left (\log \left (3 \, e^{3}\right )\right )^{2}} + 2 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x \log \left (\log \left (3 \, e^{3}\right )\right )^{2}}{x^{2} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right ) + 4 \, x^{2} - \log \left (\log \left (3 \, e^{3}\right )\right )^{2}} + \frac {2 \, {\left (x \log \left (3\right )^{4} - 8 \, x \log \left (3\right )^{3} + 24 \, x \log \left (3\right )^{2} - 32 \, x \log \left (3\right ) + 16 \, x\right )}}{\log \left (3\right )^{4} - 8 \, \log \left (3\right )^{3} + 24 \, \log \left (3\right )^{2} - 32 \, \log \left (3\right ) + 16} \]
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Time = 16.79 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x\,\left (2\,x^2\,{\ln \left (3\right )}^2-{\ln \left (\ln \left (3\right )+3\right )}^2-8\,x^2\,\ln \left (3\right )+8\,x^2\right )}{x^2\,{\ln \left (3\right )}^2-{\ln \left (\ln \left (3\right )+3\right )}^2-4\,x^2\,\ln \left (3\right )+4\,x^2} \]
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