Integrand size = 39, antiderivative size = 26 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=\log \left (\frac {x}{-3+x+\frac {x+\frac {1}{4} \left (x^8+\log \left (x^2\right )\right )}{x}}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6873, 12, 6874, 6816} \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=2 \log (x)-\log \left (-x^8-4 x^2-\log \left (x^2\right )+8 x\right ) \]
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Rule 12
Rule 6816
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (1+4 x+3 x^8-\log \left (x^2\right )\right )}{8 x^2-4 x^3-x^9-x \log \left (x^2\right )} \, dx \\ & = 2 \int \frac {1+4 x+3 x^8-\log \left (x^2\right )}{8 x^2-4 x^3-x^9-x \log \left (x^2\right )} \, dx \\ & = 2 \int \left (\frac {1}{x}+\frac {-1+4 x-4 x^2-4 x^8}{x \left (-8 x+4 x^2+x^8+\log \left (x^2\right )\right )}\right ) \, dx \\ & = 2 \log (x)+2 \int \frac {-1+4 x-4 x^2-4 x^8}{x \left (-8 x+4 x^2+x^8+\log \left (x^2\right )\right )} \, dx \\ & = 2 \log (x)-\log \left (8 x-4 x^2-x^8-\log \left (x^2\right )\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=2 \log (x)-\log \left (-8 x+4 x^2+x^8+\log \left (x^2\right )\right ) \]
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Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
method | result | size |
norman | \(\ln \left (x^{2}\right )-\ln \left (x^{8}+4 x^{2}+\ln \left (x^{2}\right )-8 x \right )\) | \(25\) |
risch | \(2 \ln \left (x \right )-\ln \left (x^{8}+4 x^{2}+\ln \left (x^{2}\right )-8 x \right )\) | \(25\) |
parallelrisch | \(\ln \left (x^{2}\right )-\ln \left (x^{8}+4 x^{2}+\ln \left (x^{2}\right )-8 x \right )\) | \(25\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=-\log \left (x^{8} + 4 \, x^{2} - 8 \, x + \log \left (x^{2}\right )\right ) + \log \left (x^{2}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=2 \log {\left (x \right )} - \log {\left (x^{8} + 4 x^{2} - 8 x + \log {\left (x^{2} \right )} \right )} \]
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Time = 0.23 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=-\log \left (\frac {1}{2} \, x^{8} + 2 \, x^{2} - 4 \, x + \log \left (x\right )\right ) + 2 \, \log \left (x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=-\log \left (x^{8} + 4 \, x^{2} - 8 \, x + \log \left (x^{2}\right )\right ) + 2 \, \log \left (x\right ) \]
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Timed out. \[ \int \frac {-2-8 x-6 x^8+2 \log \left (x^2\right )}{-8 x^2+4 x^3+x^9+x \log \left (x^2\right )} \, dx=\int -\frac {8\,x-2\,\ln \left (x^2\right )+6\,x^8+2}{x\,\ln \left (x^2\right )-8\,x^2+4\,x^3+x^9} \,d x \]
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