Integrand size = 42, antiderivative size = 20 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\log \left (8 \left (-1+\frac {25}{x^4 (4+x)^2 \log ^2(\log (4))}\right )\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(20)=40\).
Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2099, 1601} \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\log \left (x^3 (-\log (\log (4)))-4 x^2 \log (\log (4))+5\right )+\log \left (x^3 \log (\log (4))+4 x^2 \log (\log (4))+5\right )-4 \log (x)-2 \log (x+4) \]
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Rule 1601
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {4}{x}-\frac {2}{4+x}+\frac {x (8+3 x) \log (\log (4))}{-5+4 x^2 \log (\log (4))+x^3 \log (\log (4))}+\frac {x (8+3 x) \log (\log (4))}{5+4 x^2 \log (\log (4))+x^3 \log (\log (4))}\right ) \, dx \\ & = -4 \log (x)-2 \log (4+x)+\log (\log (4)) \int \frac {x (8+3 x)}{-5+4 x^2 \log (\log (4))+x^3 \log (\log (4))} \, dx+\log (\log (4)) \int \frac {x (8+3 x)}{5+4 x^2 \log (\log (4))+x^3 \log (\log (4))} \, dx \\ & = -4 \log (x)-2 \log (4+x)+\log \left (5-4 x^2 \log (\log (4))-x^3 \log (\log (4))\right )+\log \left (5+4 x^2 \log (\log (4))+x^3 \log (\log (4))\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=50 \left (-\frac {2 \log (x)}{25}-\frac {1}{25} \log (4+x)+\frac {1}{50} \log \left (25-16 x^4 \log ^2(\log (4))-8 x^5 \log ^2(\log (4))-x^6 \log ^2(\log (4))\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(20)=40\).
Time = 0.52 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.80
method | result | size |
default | \(-4 \ln \left (x \right )+\ln \left (x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )+5\right )+\ln \left (x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )-5\right )-2 \ln \left (4+x \right )\) | \(56\) |
norman | \(-4 \ln \left (x \right )+\ln \left (x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )+5\right )+\ln \left (x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )-5\right )-2 \ln \left (4+x \right )\) | \(56\) |
parallelrisch | \(-4 \ln \left (x \right )+\ln \left (\frac {x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )-5}{\ln \left (2 \ln \left (2\right )\right )}\right )+\ln \left (\frac {x^{3} \ln \left (2 \ln \left (2\right )\right )+4 x^{2} \ln \left (2 \ln \left (2\right )\right )+5}{\ln \left (2 \ln \left (2\right )\right )}\right )-2 \ln \left (4+x \right )\) | \(72\) |
risch | \(-4 \ln \left (-x \right )-2 \ln \left (4+x \right )+\ln \left (\left (-\ln \left (2\right )^{2}-2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-\ln \left (\ln \left (2\right )\right )^{2}\right ) x^{6}+\left (-8 \ln \left (2\right )^{2}-16 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-8 \ln \left (\ln \left (2\right )\right )^{2}\right ) x^{5}+\left (-16 \ln \left (2\right )^{2}-32 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )-16 \ln \left (\ln \left (2\right )\right )^{2}\right ) x^{4}+25\right )\) | \(92\) |
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none
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\log \left ({\left (x^{6} + 8 \, x^{5} + 16 \, x^{4}\right )} \log \left (2 \, \log \left (2\right )\right )^{2} - 25\right ) - 2 \, \log \left (x + 4\right ) - 4 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 10.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=- 4 \log {\left (x \right )} - 2 \log {\left (x + 4 \right )} + \log {\left (x^{6} + 8 x^{5} + 16 x^{4} - \frac {25}{2 \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )} + \log {\left (\log {\left (2 \right )} \right )}^{2} + \log {\left (2 \right )}^{2}} \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.75 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\log \left (x^{3} \log \left (2 \, \log \left (2\right )\right ) + 4 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) + 5\right ) + \log \left (x^{3} \log \left (2 \, \log \left (2\right )\right ) + 4 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) - 5\right ) - 2 \, \log \left (x + 4\right ) - 4 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.95 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\log \left ({\left | x^{3} \log \left (2 \, \log \left (2\right )\right ) + 4 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) + 5 \right |}\right ) + \log \left ({\left | x^{3} \log \left (2 \, \log \left (2\right )\right ) + 4 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) - 5 \right |}\right ) - 2 \, \log \left ({\left | x + 4 \right |}\right ) - 4 \, \log \left ({\left | x \right |}\right ) \]
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Time = 13.89 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.15 \[ \int \frac {400+150 x}{-100 x-25 x^2+\left (64 x^5+48 x^6+12 x^7+x^8\right ) \log ^2(\log (4))} \, dx=\ln \left ({\ln \left (\ln \left (4\right )\right )}^2\,x^6+8\,{\ln \left (\ln \left (4\right )\right )}^2\,x^5+16\,{\ln \left (\ln \left (4\right )\right )}^2\,x^4-25\right )-2\,\ln \left (x+4\right )-4\,\ln \left (x\right ) \]
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