Integrand size = 16, antiderivative size = 15 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=\log \left (\frac {-4+2 x}{16 (4+x)^4}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {646, 31} \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=\log (2-x)-4 \log (x+4) \]
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Rule 31
Rule 646
Rubi steps \begin{align*} \text {integral}& = -\left (4 \int \frac {1}{4+x} \, dx\right )+\int \frac {1}{-2+x} \, dx \\ & = \log (2-x)-4 \log (4+x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.40 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=-3 \left (-\frac {1}{3} \log (2-x)+\frac {4}{3} \log (4+x)\right ) \]
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Time = 0.74 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80
method | result | size |
default | \(\ln \left (-2+x \right )-4 \ln \left (4+x \right )\) | \(12\) |
norman | \(\ln \left (-2+x \right )-4 \ln \left (4+x \right )\) | \(12\) |
risch | \(\ln \left (-2+x \right )-4 \ln \left (4+x \right )\) | \(12\) |
parallelrisch | \(\ln \left (-2+x \right )-4 \ln \left (4+x \right )\) | \(12\) |
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none
Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=-4 \, \log \left (x + 4\right ) + \log \left (x - 2\right ) \]
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Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=\log {\left (x - 2 \right )} - 4 \log {\left (x + 4 \right )} \]
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none
Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=-4 \, \log \left (x + 4\right ) + \log \left (x - 2\right ) \]
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none
Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=-4 \, \log \left ({\left | x + 4 \right |}\right ) + \log \left ({\left | x - 2 \right |}\right ) \]
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Time = 11.92 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \frac {12-3 x}{-8+2 x+x^2} \, dx=\ln \left (x-2\right )-4\,\ln \left (x+4\right ) \]
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