Optimal. Leaf size=12 \[ -\log (x)+\log (1-b x) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {36, 29, 31}
\begin {gather*} \log (1-b x)-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {1}{x (-1+b x)} \, dx &=b \int \frac {1}{-1+b x} \, dx-\int \frac {1}{x} \, dx\\ &=-\log (x)+\log (1-b x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\log (x)+\log (1-b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 12, normalized size = 1.00
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (b x -1\right )\) | \(12\) |
norman | \(-\ln \left (x \right )+\ln \left (b x -1\right )\) | \(12\) |
risch | \(-\ln \left (x \right )+\ln \left (-b x +1\right )\) | \(13\) |
meijerg | \(-\ln \left (x \right )-\ln \left (-b \right )+\ln \left (-b x +1\right )\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 11, normalized size = 0.92 \begin {gather*} \log \left (b x - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.49, size = 11, normalized size = 0.92 \begin {gather*} \log \left (b x - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} - \log {\left (x \right )} + \log {\left (x - \frac {1}{b} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 13, normalized size = 1.08 \begin {gather*} \log \left ({\left | b x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 9, normalized size = 0.75 \begin {gather*} -2\,\mathrm {atanh}\left (2\,b\,x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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