Integrand size = 188, antiderivative size = 32 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \left (3 x-\log \left (2 x+\frac {2 x^2}{-\frac {5}{2}+x^2}\right )\right )} \]
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Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6820, 12, 6843, 32} \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=-\frac {1}{15 \left (1-\frac {3 x}{\log \left (\frac {2 x \left (-2 x^2-2 x+5\right )}{5-2 x^2}\right )}\right )} \]
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Rule 12
Rule 32
Rule 6820
Rule 6843
Rubi steps \begin{align*} \text {integral}& = \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}{5 \left (25-10 x-20 x^2+4 x^3+4 x^4\right ) \left (3 x-\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )^2} \, dx \\ & = \frac {1}{5} \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}{\left (25-10 x-20 x^2+4 x^3+4 x^4\right ) \left (3 x-\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )^2} \, dx \\ & = -\left (\frac {1}{5} \text {Subst}\left (\int \frac {1}{(-1+3 x)^2} \, dx,x,\frac {x}{\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )}\right )\right ) \\ & = -\frac {1}{15 \left (1-\frac {3 x}{\log \left (\frac {2 x \left (5-2 x-2 x^2\right )}{5-2 x^2}\right )}\right )} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=-\frac {x}{5 \left (-3 x+\log \left (\frac {2 x \left (-5+2 x+2 x^2\right )}{-5+2 x^2}\right )\right )} \]
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Time = 0.47 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16
method | result | size |
risch | \(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\) | \(37\) |
parallelrisch | \(\frac {x}{15 x -5 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\) | \(37\) |
norman | \(\frac {\ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}{45 x -15 \ln \left (\frac {4 x^{3}+4 x^{2}-10 x}{2 x^{2}-5}\right )}\) | \(61\) |
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Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=- \frac {x}{- 15 x + 5 \log {\left (\frac {4 x^{3} + 4 x^{2} - 10 x}{2 x^{2} - 5} \right )}} \]
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Time = 0.36 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (2\right ) - \log \left (2 \, x^{2} + 2 \, x - 5\right ) + \log \left (2 \, x^{2} - 5\right ) - \log \left (x\right )\right )}} \]
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Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5 \, {\left (3 \, x - \log \left (\frac {2 \, {\left (2 \, x^{3} + 2 \, x^{2} - 5 \, x\right )}}{2 \, x^{2} - 5}\right )\right )}} \]
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Time = 10.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {25-20 x-20 x^2+4 x^4+\left (-25+10 x+20 x^2-4 x^3-4 x^4\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )}{1125 x^2-450 x^3-900 x^4+180 x^5+180 x^6+\left (-750 x+300 x^2+600 x^3-120 x^4-120 x^5\right ) \log \left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )+\left (125-50 x-100 x^2+20 x^3+20 x^4\right ) \log ^2\left (\frac {-10 x+4 x^2+4 x^3}{-5+2 x^2}\right )} \, dx=\frac {x}{5\,\left (3\,x-\ln \left (\frac {4\,x^3+4\,x^2-10\,x}{2\,x^2-5}\right )\right )} \]
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