Integrand size = 35, antiderivative size = 25 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=\log \left (\frac {4}{x (2+x) \left (2+x+x^2\right )^2 (-3+\log (5))}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2099, 642} \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-2 \log \left (x^2+x+2\right )-\log (x)-\log (x+2) \]
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Rule 642
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{-2-x}-\frac {1}{x}-\frac {2 (1+2 x)}{2+x+x^2}\right ) \, dx \\ & = -\log (x)-\log (2+x)-2 \int \frac {1+2 x}{2+x+x^2} \, dx \\ & = -\log (x)-\log (2+x)-2 \log \left (2+x+x^2\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-2 \left (\frac {\log (x)}{2}+\frac {1}{2} \log (2+x)+\log \left (2+x+x^2\right )\right ) \]
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Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\ln \left (2+x \right )-2 \ln \left (x^{2}+x +2\right )-\ln \left (x \right )\) | \(21\) |
norman | \(-\ln \left (2+x \right )-2 \ln \left (x^{2}+x +2\right )-\ln \left (x \right )\) | \(21\) |
risch | \(-2 \ln \left (x^{2}+x +2\right )-\ln \left (x^{2}+2 x \right )\) | \(21\) |
parallelrisch | \(-\ln \left (2+x \right )-2 \ln \left (x^{2}+x +2\right )-\ln \left (x \right )\) | \(21\) |
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none
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-\log \left (x^{2} + 2 \, x\right ) - 2 \, \log \left (x^{2} + x + 2\right ) \]
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Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=- \log {\left (x^{2} + 2 x \right )} - 2 \log {\left (x^{2} + x + 2 \right )} \]
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none
Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-2 \, \log \left (x^{2} + x + 2\right ) - \log \left (x + 2\right ) - \log \left (x\right ) \]
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-2 \, \log \left (x^{2} + x + 2\right ) - \log \left ({\left | x + 2 \right |}\right ) - \log \left ({\left | x \right |}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {-4-10 x-14 x^2-6 x^3}{4 x+4 x^2+3 x^3+x^4} \, dx=-\ln \left (x\,\left (x+2\right )\right )-2\,\ln \left (x^2+x+2\right ) \]
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