Integrand size = 35, antiderivative size = 21 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-8+\log \left (e^{-x} \left (3-\frac {16}{x^2}-x^4\right )\right ) \]
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Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1608, 6874, 1601} \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=\log \left (x^6-3 x^2+16\right )-x-2 \log (x) \]
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Rule 1601
Rule 1608
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{x \left (16-3 x^2+x^6\right )} \, dx \\ & = \int \left (-1-\frac {2}{x}+\frac {6 x \left (-1+x^4\right )}{16-3 x^2+x^6}\right ) \, dx \\ & = -x-2 \log (x)+6 \int \frac {x \left (-1+x^4\right )}{16-3 x^2+x^6} \, dx \\ & = -x-2 \log (x)+\log \left (16-3 x^2+x^6\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x-2 \log (x)+\log \left (16-3 x^2+x^6\right ) \]
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Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
default | \(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\) | \(20\) |
norman | \(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\) | \(20\) |
risch | \(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\) | \(20\) |
parallelrisch | \(-x +\ln \left (x^{6}-3 x^{2}+16\right )-2 \ln \left (x \right )\) | \(20\) |
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none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=- x - 2 \log {\left (x \right )} + \log {\left (x^{6} - 3 x^{2} + 16 \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left (x\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=-x + \log \left (x^{6} - 3 \, x^{2} + 16\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]
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Time = 13.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {-32-16 x+3 x^3+4 x^6-x^7}{16 x-3 x^3+x^7} \, dx=\ln \left (x^6-3\,x^2+16\right )-x-2\,\ln \left (x\right ) \]
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