Integrand size = 18, antiderivative size = 22 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{(3+2 x) \log (5)}+\log \left (4+e^{e^4}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 27, 32} \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{(2 x+3) \log (5)} \]
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Rule 12
Rule 27
Rule 32
Rubi steps \begin{align*} \text {integral}& = \frac {4 \int \frac {1}{9+12 x+4 x^2} \, dx}{\log (5)} \\ & = \frac {4 \int \frac {1}{(3+2 x)^2} \, dx}{\log (5)} \\ & = -\frac {2}{(3+2 x) \log (5)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{(3+2 x) \log (5)} \]
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Time = 0.34 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55
method | result | size |
risch | \(-\frac {1}{\ln \left (5\right ) \left (x +\frac {3}{2}\right )}\) | \(12\) |
gosper | \(-\frac {2}{\left (3+2 x \right ) \ln \left (5\right )}\) | \(14\) |
default | \(-\frac {2}{\left (3+2 x \right ) \ln \left (5\right )}\) | \(14\) |
norman | \(-\frac {2}{\left (3+2 x \right ) \ln \left (5\right )}\) | \(14\) |
parallelrisch | \(-\frac {2}{\left (3+2 x \right ) \ln \left (5\right )}\) | \(14\) |
meijerg | \(\frac {4 x}{9 \ln \left (5\right ) \left (1+\frac {2 x}{3}\right )}\) | \(15\) |
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Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{{\left (2 \, x + 3\right )} \log \left (5\right )} \]
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Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=- \frac {4}{4 x \log {\left (5 \right )} + 6 \log {\left (5 \right )}} \]
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Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{{\left (2 \, x + 3\right )} \log \left (5\right )} \]
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Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{{\left (2 \, x + 3\right )} \log \left (5\right )} \]
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Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.59 \[ \int \frac {4}{\left (9+12 x+4 x^2\right ) \log (5)} \, dx=-\frac {2}{\ln \left (5\right )\,\left (2\,x+3\right )} \]
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