Integrand size = 85, antiderivative size = 24 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\frac {-\frac {1}{x}+x}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \]
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\[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{x^4 \left (-20+10 x+x^2\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx \\ & = \int \left (-\frac {2 \left (-5-x+5 x^2+x^3\right )}{x^3 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {3-x^2}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {-5-x+5 x^2+x^3}{x^3 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx\right )+\int \frac {3-x^2}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ & = -\left (2 \int \left (\frac {1}{4 x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {7}{40 x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )}-\frac {3}{20 x \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {3 (31+2 x)}{40 \left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\int \left (\frac {3}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )}-\frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx \\ & = -\left (\frac {3}{20} \int \frac {31+2 x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ & = -\left (\frac {3}{20} \int \left (\frac {31}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {2 x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ & = \frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {3}{10} \int \frac {x}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {93}{20} \int \frac {1}{\left (-20+10 x+x^2\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ & = -\left (\frac {3}{10} \int \left (\frac {1-\frac {\sqrt {5}}{3}}{\left (10-6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}+\frac {1+\frac {\sqrt {5}}{3}}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx\right )+\frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {93}{20} \int \left (-\frac {1}{3 \sqrt {5} \left (-10+6 \sqrt {5}-2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}-\frac {1}{3 \sqrt {5} \left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )}\right ) \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ & = \frac {3}{10} \int \frac {1}{x \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {7}{20} \int \frac {1}{x^2 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{2} \int \frac {1}{x^3 \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx+3 \int \frac {1}{x^4 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx+\frac {31 \int \frac {1}{\left (-10+6 \sqrt {5}-2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx}{20 \sqrt {5}}+\frac {31 \int \frac {1}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx}{20 \sqrt {5}}-\frac {1}{10} \left (3-\sqrt {5}\right ) \int \frac {1}{\left (10-6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\frac {1}{10} \left (3+\sqrt {5}\right ) \int \frac {1}{\left (10+6 \sqrt {5}+2 x\right ) \log ^2\left (-2+x+\frac {x^2}{10}\right )} \, dx-\int \frac {1}{x^2 \log \left (-2+x+\frac {x^2}{10}\right )} \, dx \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=-\frac {1-x^2}{x^3 \log \left (-2+x+\frac {x^2}{10}\right )} \]
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Time = 2.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
norman | \(\frac {x^{2}-1}{\ln \left (\frac {1}{10} x^{2}+x -2\right ) x^{3}}\) | \(21\) |
risch | \(\frac {x^{2}-1}{\ln \left (\frac {1}{10} x^{2}+x -2\right ) x^{3}}\) | \(21\) |
parallelrisch | \(\frac {x^{2}-1}{\ln \left (\frac {1}{10} x^{2}+x -2\right ) x^{3}}\) | \(21\) |
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Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\frac {x^{2} - 1}{x^{3} \log \left (\frac {1}{10} \, x^{2} + x - 2\right )} \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\frac {x^{2} - 1}{x^{3} \log {\left (\frac {x^{2}}{10} + x - 2 \right )}} \]
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Time = 0.31 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=-\frac {x^{2} - 1}{x^{3} {\left (\log \left (5\right ) + \log \left (2\right )\right )} - x^{3} \log \left (x^{2} + 10 \, x - 20\right )} \]
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Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\frac {x^{2} - 1}{x^{3} \log \left (\frac {1}{10} \, x^{2} + x - 2\right )} \]
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Time = 12.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {10 x+2 x^2-10 x^3-2 x^4+\left (-60+30 x+23 x^2-10 x^3-x^4\right ) \log \left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )}{\left (-20 x^4+10 x^5+x^6\right ) \log ^2\left (\frac {1}{10} \left (-20+10 x+x^2\right )\right )} \, dx=\frac {x^2-1}{x^3\,\ln \left (\frac {x^2}{10}+x-2\right )} \]
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