Integrand size = 30, antiderivative size = 23 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=81+2 \left (1+e^4\right )+\log \left (\left (3+\frac {1}{2 x}\right )^2+x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2099, 1601} \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=\log \left (4 x^3+36 x^2+12 x+1\right )-2 \log (x) \]
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Rule 1601
Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{x}+\frac {12 \left (1+6 x+x^2\right )}{1+12 x+36 x^2+4 x^3}\right ) \, dx \\ & = -2 \log (x)+12 \int \frac {1+6 x+x^2}{1+12 x+36 x^2+4 x^3} \, dx \\ & = -2 \log (x)+\log \left (1+12 x+36 x^2+4 x^3\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=-2 \log (x)+\log \left (1+12 x+36 x^2+4 x^3\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
parallelrisch | \(-2 \ln \left (x \right )+\ln \left (x^{3}+9 x^{2}+3 x +\frac {1}{4}\right )\) | \(20\) |
default | \(-2 \ln \left (x \right )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\) | \(22\) |
norman | \(-2 \ln \left (x \right )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\) | \(22\) |
risch | \(-2 \ln \left (x \right )+\ln \left (4 x^{3}+36 x^{2}+12 x +1\right )\) | \(22\) |
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Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=\log \left (4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1\right ) - 2 \, \log \left (x\right ) \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=- 2 \log {\left (x \right )} + \log {\left (4 x^{3} + 36 x^{2} + 12 x + 1 \right )} \]
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Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=\log \left (4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1\right ) - 2 \, \log \left (x\right ) \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=\log \left ({\left | 4 \, x^{3} + 36 \, x^{2} + 12 \, x + 1 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right ) \]
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Time = 8.59 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {-2-12 x+4 x^3}{x+12 x^2+36 x^3+4 x^4} \, dx=\ln \left (x^3+9\,x^2+3\,x+\frac {1}{4}\right )-2\,\ln \left (x\right ) \]
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