Integrand size = 124, antiderivative size = 23 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x+\left (-5+2 x+x^2+\log ^2(x)+\log ^2\left (\log ^4(x)\right )\right )^2 \]
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\[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=\int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-19 x-12 x^2+12 x^3+4 x^4-20 \log (x)+8 x \log (x)+4 x^2 \log (x)+4 x \log ^2(x)+4 x^2 \log ^2(x)+4 \log ^3(x)}{x}+\frac {16 \left (-5+2 x+x^2+\log ^2(x)\right ) \log \left (\log ^4(x)\right )}{x \log (x)}+\frac {4 \left (x+x^2+\log (x)\right ) \log ^2\left (\log ^4(x)\right )}{x}+\frac {16 \log ^3\left (\log ^4(x)\right )}{x \log (x)}\right ) \, dx \\ & = 4 \int \frac {\left (x+x^2+\log (x)\right ) \log ^2\left (\log ^4(x)\right )}{x} \, dx+16 \int \frac {\left (-5+2 x+x^2+\log ^2(x)\right ) \log \left (\log ^4(x)\right )}{x \log (x)} \, dx+16 \int \frac {\log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx+\int \frac {-19 x-12 x^2+12 x^3+4 x^4-20 \log (x)+8 x \log (x)+4 x^2 \log (x)+4 x \log ^2(x)+4 x^2 \log ^2(x)+4 \log ^3(x)}{x} \, dx \\ & = 4 \int \left (\log ^2\left (\log ^4(x)\right )+x \log ^2\left (\log ^4(x)\right )+\frac {\log (x) \log ^2\left (\log ^4(x)\right )}{x}\right ) \, dx+16 \int \left (\frac {2 \log \left (\log ^4(x)\right )}{\log (x)}-\frac {5 \log \left (\log ^4(x)\right )}{x \log (x)}+\frac {x \log \left (\log ^4(x)\right )}{\log (x)}+\frac {\log (x) \log \left (\log ^4(x)\right )}{x}\right ) \, dx+16 \text {Subst}\left (\int \frac {\log ^3\left (x^4\right )}{x} \, dx,x,\log (x)\right )+\int \left (-19-12 x+12 x^2+4 x^3+\frac {4 \left (-5+2 x+x^2\right ) \log (x)}{x}+4 (1+x) \log ^2(x)+\frac {4 \log ^3(x)}{x}\right ) \, dx \\ & = -19 x-6 x^2+4 x^3+x^4+4 \int \frac {\left (-5+2 x+x^2\right ) \log (x)}{x} \, dx+4 \int (1+x) \log ^2(x) \, dx+4 \int \frac {\log ^3(x)}{x} \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+4 \int \frac {\log (x) \log ^2\left (\log ^4(x)\right )}{x} \, dx+4 \text {Subst}\left (\int x^3 \, dx,x,\log \left (\log ^4(x)\right )\right )+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+16 \int \frac {\log (x) \log \left (\log ^4(x)\right )}{x} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx-80 \int \frac {\log \left (\log ^4(x)\right )}{x \log (x)} \, dx \\ & = -19 x-6 x^2+4 x^3+x^4+\log ^4\left (\log ^4(x)\right )+4 \int \left (2 \log (x)-\frac {5 \log (x)}{x}+x \log (x)\right ) \, dx+4 \int \left (\log ^2(x)+x \log ^2(x)\right ) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+4 \text {Subst}\left (\int x^3 \, dx,x,\log (x)\right )+4 \text {Subst}\left (\int x \log ^2\left (x^4\right ) \, dx,x,\log (x)\right )+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+16 \text {Subst}\left (\int x \log \left (x^4\right ) \, dx,x,\log (x)\right )+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx-80 \text {Subst}\left (\int \frac {\log \left (x^4\right )}{x} \, dx,x,\log (x)\right ) \\ & = -19 x-6 x^2+4 x^3+x^4-16 \log ^2(x)+\log ^4(x)+8 \log ^2(x) \log \left (\log ^4(x)\right )-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )+4 \int x \log (x) \, dx+4 \int \log ^2(x) \, dx+4 \int x \log ^2(x) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+8 \int \log (x) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx-16 \text {Subst}\left (\int x \log \left (x^4\right ) \, dx,x,\log (x)\right )-20 \int \frac {\log (x)}{x} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx \\ & = -27 x-7 x^2+4 x^3+x^4+8 x \log (x)+2 x^2 \log (x)-10 \log ^2(x)+4 x \log ^2(x)+2 x^2 \log ^2(x)+\log ^4(x)-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )-4 \int x \log (x) \, dx+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx-8 \int \log (x) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx \\ & = -19 x-6 x^2+4 x^3+x^4-10 \log ^2(x)+4 x \log ^2(x)+2 x^2 \log ^2(x)+\log ^4(x)-10 \log ^2\left (\log ^4(x)\right )+2 \log ^2(x) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right )+4 \int \log ^2\left (\log ^4(x)\right ) \, dx+4 \int x \log ^2\left (\log ^4(x)\right ) \, dx+16 \int \frac {x \log \left (\log ^4(x)\right )}{\log (x)} \, dx+32 \int \frac {\log \left (\log ^4(x)\right )}{\log (x)} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(23)=46\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=-19 x-6 x^2+4 x^3+x^4+2 \left (-5+2 x+x^2\right ) \log ^2(x)+\log ^4(x)+2 \left (-5+2 x+x^2+\log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+\log ^4\left (\log ^4(x)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(23)=46\).
Time = 1.96 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.13
method | result | size |
parallelrisch | \(x^{4}+2 x^{2} \ln \left (x \right )^{2}+2 \ln \left (\ln \left (x \right )^{4}\right )^{2} x^{2}+\ln \left (x \right )^{4}+2 \ln \left (x \right )^{2} \ln \left (\ln \left (x \right )^{4}\right )^{2}+\ln \left (\ln \left (x \right )^{4}\right )^{4}+4 x^{3}+4 x \ln \left (x \right )^{2}+4 x \ln \left (\ln \left (x \right )^{4}\right )^{2}-6 x^{2}-10 \ln \left (x \right )^{2}-10 \ln \left (\ln \left (x \right )^{4}\right )^{2}-19 x\) | \(95\) |
risch | \(\text {Expression too large to display}\) | \(12852\) |
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (\log \left (x\right )^{4}\right )^{4} + \log \left (x\right )^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \left (x\right )^{2} - 6 \, x^{2} - 19 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + 4 x^{3} - 6 x^{2} - 19 x + \left (2 x^{2} + 4 x - 10\right ) \log {\left (x \right )}^{2} + \left (2 x^{2} + 4 x + 2 \log {\left (x \right )}^{2} - 10\right ) \log {\left (\log {\left (x \right )}^{4} \right )}^{2} + \log {\left (x \right )}^{4} + \log {\left (\log {\left (x \right )}^{4} \right )}^{4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 94, normalized size of antiderivative = 4.09 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (x\right )^{4} + 256 \, \log \left (\log \left (x\right )\right )^{4} + {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + 4 \, x^{3} + 2 \, x^{2} \log \left (x\right ) + 32 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} + 4 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - 7 \, x^{2} + 8 \, x \log \left (x\right ) - 10 \, \log \left (x\right )^{2} - 27 \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (23) = 46\).
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx=x^{4} + \log \left (\log \left (x\right )^{4}\right )^{4} + \log \left (x\right )^{4} + 4 \, x^{3} + 2 \, {\left (x^{2} + \log \left (x\right )^{2} + 2 \, x - 5\right )} \log \left (\log \left (x\right )^{4}\right )^{2} + 2 \, {\left (x^{2} + 2 \, x - 5\right )} \log \left (x\right )^{2} - 6 \, x^{2} - 19 \, x \]
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Time = 12.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {\left (-19 x-12 x^2+12 x^3+4 x^4\right ) \log (x)+\left (-20+8 x+4 x^2\right ) \log ^2(x)+\left (4 x+4 x^2\right ) \log ^3(x)+4 \log ^4(x)+\left (-80+32 x+16 x^2+16 \log ^2(x)\right ) \log \left (\log ^4(x)\right )+\left (\left (4 x+4 x^2\right ) \log (x)+4 \log ^2(x)\right ) \log ^2\left (\log ^4(x)\right )+16 \log ^3\left (\log ^4(x)\right )}{x \log (x)} \, dx={\ln \left ({\ln \left (x\right )}^4\right )}^4-19\,x+{\ln \left (x\right )}^2\,\left (2\,x^2+4\,x-10\right )+{\ln \left (x\right )}^4+{\ln \left ({\ln \left (x\right )}^4\right )}^2\,\left (2\,x^2+4\,x+2\,{\ln \left (x\right )}^2-10\right )-6\,x^2+4\,x^3+x^4 \]
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