Integrand size = 11, antiderivative size = 18 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=\frac {1}{x}-b \log (x)+b \log (1-b x) \]
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Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46} \[ \int \frac {1}{x^2 (-1+b x)} \, dx=-b \log (x)+b \log (1-b x)+\frac {1}{x} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^2}-\frac {b}{x}+\frac {b^2}{-1+b x}\right ) \, dx \\ & = \frac {1}{x}-b \log (x)+b \log (1-b x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=\frac {1}{x}-b \log (x)+b \log (1-b x) \]
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Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {1}{x}-b \ln \left (x \right )+b \ln \left (b x -1\right )\) | \(18\) |
norman | \(\frac {1}{x}-b \ln \left (x \right )+b \ln \left (b x -1\right )\) | \(18\) |
risch | \(\frac {1}{x}-b \ln \left (x \right )+b \ln \left (-b x +1\right )\) | \(19\) |
parallelrisch | \(-\frac {b \ln \left (x \right ) x -b \ln \left (b x -1\right ) x -1}{x}\) | \(23\) |
meijerg | \(b \left (\frac {1}{x b}-\ln \left (x \right )-\ln \left (-b \right )+\ln \left (-b x +1\right )\right )\) | \(28\) |
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none
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=\frac {b x \log \left (b x - 1\right ) - b x \log \left (x\right ) + 1}{x} \]
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Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=b \left (- \log {\left (x \right )} + \log {\left (x - \frac {1}{b} \right )}\right ) + \frac {1}{x} \]
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none
Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=b \log \left (b x - 1\right ) - b \log \left (x\right ) + \frac {1}{x} \]
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=b \log \left ({\left | b x - 1 \right |}\right ) - b \log \left ({\left | x \right |}\right ) + \frac {1}{x} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 (-1+b x)} \, dx=\frac {1}{x}-2\,b\,\mathrm {atanh}\left (2\,b\,x-1\right ) \]
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