Integrand size = 127, antiderivative size = 26 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {x}{\frac {1}{5} x (5+x)+5 \log (x) (-5-\log (3)+\log (\log (x)))} \]
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\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))\right )}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx \\ & = 5 \int \frac {-x^2+100 \left (1+\frac {\log (3)}{4}\right )-25 \log (x) (5+\log (3)-\log (\log (x)))-25 \log (\log (x))}{\left (x (5+x)+25 \log (x) \left (-5+\log \left (\frac {\log (x)}{3}\right )\right )\right )^2} \, dx \\ & = 5 \int \left (-\frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (4+\log (3))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}-\frac {125 \left (1+\frac {\log (3)}{5}\right ) \log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {25 (-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \frac {(-1+\log (x)) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )+125 \int \left (-\frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}+\frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2}\right ) \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {x^2}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx\right )-125 \int \frac {\log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+125 \int \frac {\log (x) \log (\log (x))}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx+(125 (4+\log (3))) \int \frac {1}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx-(125 (5+\log (3))) \int \frac {\log (x)}{\left (5 x+x^2-125 \log (x)+25 \log (x) \log \left (\frac {\log (x)}{3}\right )\right )^2} \, dx \\ \end{align*}
\[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx \]
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Time = 0.90 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
default | \(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\) | \(32\) |
risch | \(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\) | \(32\) |
parallelrisch | \(-\frac {5 x}{25 \ln \left (3\right ) \ln \left (x \right )-25 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )-x^{2}+125 \ln \left (x \right )-5 x}\) | \(32\) |
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Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, {\left (\log \left (3\right ) + 5\right )} \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x} \]
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Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 x}{x^{2} + 5 x + 25 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )} - 125 \log {\left (x \right )} - 25 \log {\left (3 \right )} \log {\left (x \right )}} \]
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Time = 0.30 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, {\left (\log \left (3\right ) + 5\right )} \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x} \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\frac {5 \, x}{x^{2} - 25 \, \log \left (3\right ) \log \left (x\right ) + 25 \, \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 5 \, x - 125 \, \log \left (x\right )} \]
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Timed out. \[ \int \frac {500-5 x^2+125 \log (3)+(-625-125 \log (3)) \log (x)+(-125+125 \log (x)) \log (\log (x))}{25 x^2+10 x^3+x^4+\left (-1250 x-250 x^2+\left (-250 x-50 x^2\right ) \log (3)\right ) \log (x)+\left (15625+6250 \log (3)+625 \log ^2(3)\right ) \log ^2(x)+\left (\left (250 x+50 x^2\right ) \log (x)+(-6250-1250 \log (3)) \log ^2(x)\right ) \log (\log (x))+625 \log ^2(x) \log ^2(\log (x))} \, dx=\int \frac {125\,\ln \left (3\right )+\ln \left (\ln \left (x\right )\right )\,\left (125\,\ln \left (x\right )-125\right )-\ln \left (x\right )\,\left (125\,\ln \left (3\right )+625\right )-5\,x^2+500}{625\,{\ln \left (\ln \left (x\right )\right )}^2\,{\ln \left (x\right )}^2+\ln \left (\ln \left (x\right )\right )\,\left (\ln \left (x\right )\,\left (50\,x^2+250\,x\right )-{\ln \left (x\right )}^2\,\left (1250\,\ln \left (3\right )+6250\right )\right )+25\,x^2+10\,x^3+x^4-\ln \left (x\right )\,\left (1250\,x+\ln \left (3\right )\,\left (50\,x^2+250\,x\right )+250\,x^2\right )+{\ln \left (x\right )}^2\,\left (6250\,\ln \left (3\right )+625\,{\ln \left (3\right )}^2+15625\right )} \,d x \]
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