Integrand size = 11, antiderivative size = 15 \[ \int \frac {1}{x (a+b x)} \, dx=-\frac {\log \left (\frac {a+b x}{x}\right )}{a} \]
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Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {36, 29, 31} \[ \int \frac {1}{x (a+b x)} \, dx=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
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Rule 29
Rule 31
Rule 36
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{x} \, dx}{a}-\frac {b \int \frac {1}{a+b x} \, dx}{a} \\ & = \frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x (a+b x)} \, dx=\frac {\log (x)}{a}-\frac {\log (a+b x)}{a} \]
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Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
parallelrisch | \(\frac {\ln \left (x \right )-\ln \left (b x +a \right )}{a}\) | \(16\) |
default | \(-\frac {\ln \left (b x +a \right )}{a}+\frac {\ln \left (x \right )}{a}\) | \(19\) |
norman | \(-\frac {\ln \left (b x +a \right )}{a}+\frac {\ln \left (x \right )}{a}\) | \(19\) |
risch | \(-\frac {\ln \left (b x +a \right )}{a}+\frac {\ln \left (-x \right )}{a}\) | \(21\) |
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none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x (a+b x)} \, dx=-\frac {\log \left (b x + a\right ) - \log \left (x\right )}{a} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x (a+b x)} \, dx=\frac {\log {\left (x \right )} - \log {\left (\frac {a}{b} + x \right )}}{a} \]
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none
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x (a+b x)} \, dx=-\frac {\log \left (b x + a\right )}{a} + \frac {\log \left (x\right )}{a} \]
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none
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x (a+b x)} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a} + \frac {\log \left ({\left | x \right |}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (a+b x)} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a} \]
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