Integrand size = 36, antiderivative size = 22 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {2+4 x+(1-\log (x))^2}{\frac {1}{5}+\log (x)} \]
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Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6873, 6874, 2395, 2334, 2336, 2209, 2339, 30} \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {20 x}{5 \log (x)+1}+\log (x)+\frac {86}{5 (5 \log (x)+1)} \]
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Rule 30
Rule 2209
Rule 2334
Rule 2336
Rule 2339
Rule 2395
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x (1+5 \log (x))^2} \, dx \\ & = \int \left (\frac {1}{x}-\frac {2 (43+50 x)}{x (1+5 \log (x))^2}+\frac {20}{1+5 \log (x)}\right ) \, dx \\ & = \log (x)-2 \int \frac {43+50 x}{x (1+5 \log (x))^2} \, dx+20 \int \frac {1}{1+5 \log (x)} \, dx \\ & = \log (x)-2 \int \left (\frac {50}{(1+5 \log (x))^2}+\frac {43}{x (1+5 \log (x))^2}\right ) \, dx+20 \text {Subst}\left (\int \frac {e^x}{1+5 x} \, dx,x,\log (x)\right ) \\ & = \frac {4 \text {Ei}\left (\frac {1}{5} (1+5 \log (x))\right )}{\sqrt [5]{e}}+\log (x)-86 \int \frac {1}{x (1+5 \log (x))^2} \, dx-100 \int \frac {1}{(1+5 \log (x))^2} \, dx \\ & = \frac {4 \text {Ei}\left (\frac {1}{5} (1+5 \log (x))\right )}{\sqrt [5]{e}}+\log (x)+\frac {20 x}{1+5 \log (x)}-\frac {86}{5} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,1+5 \log (x)\right )-20 \int \frac {1}{1+5 \log (x)} \, dx \\ & = \frac {4 \text {Ei}\left (\frac {1}{5} (1+5 \log (x))\right )}{\sqrt [5]{e}}+\log (x)+\frac {86}{5 (1+5 \log (x))}+\frac {20 x}{1+5 \log (x)}-20 \text {Subst}\left (\int \frac {e^x}{1+5 x} \, dx,x,\log (x)\right ) \\ & = \log (x)+\frac {86}{5 (1+5 \log (x))}+\frac {20 x}{1+5 \log (x)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=5 \left (\frac {\log (x)}{5}+\frac {86+100 x}{25+125 \log (x)}\right ) \]
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Time = 2.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\ln \left (x \right )+\frac {20 x +\frac {86}{5}}{1+5 \ln \left (x \right )}\) | \(19\) |
norman | \(\frac {5 \ln \left (x \right )^{2}+20 x +17}{1+5 \ln \left (x \right )}\) | \(21\) |
parallelrisch | \(\frac {85+25 \ln \left (x \right )^{2}+100 x}{5+25 \ln \left (x \right )}\) | \(22\) |
default | \(\frac {86}{5 \left (1+5 \ln \left (x \right )\right )}+\ln \left (x \right )+\frac {4 x}{\frac {1}{5}+\ln \left (x \right )}\) | \(23\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {25 \, \log \left (x\right )^{2} + 100 \, x + 5 \, \log \left (x\right ) + 86}{5 \, {\left (5 \, \log \left (x\right ) + 1\right )}} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {100 x + 86}{25 \log {\left (x \right )} + 5} + \log {\left (x \right )} \]
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Time = 0.21 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {100 \, x + 1}{5 \, {\left (5 \, \log \left (x\right ) + 1\right )}} + \frac {17}{5 \, \log \left (x\right ) + 1} + \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\frac {2 \, {\left (50 \, x + 43\right )}}{5 \, {\left (5 \, \log \left (x\right ) + 1\right )}} + \log \left (x\right ) \]
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Time = 10.98 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {-85-80 x+(10+100 x) \log (x)+25 \log ^2(x)}{x+10 x \log (x)+25 x \log ^2(x)} \, dx=\ln \left (x\right )+\frac {20\,x+\frac {86}{5}}{5\,\ln \left (x\right )+1} \]
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