Optimal. Leaf size=18 \[ -b \log (x)+b \log (1-b x)+\frac {1}{x} \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {44} \begin {gather*} -b \log (x)+b \log (1-b x)+\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x^2 (-1+b x)} \, dx &=\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*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -b \log (x)+b \log (1-b x)+\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 (-1+b x)} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.73, size = 21, normalized size = 1.17 \begin {gather*} \frac {b x \log \left (b x - 1\right ) - b x \log \relax (x) + 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 19, normalized size = 1.06 \begin {gather*} b \log \left ({\left | b x - 1 \right |}\right ) - b \log \left ({\left | x \right |}\right ) + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -b \ln \relax (x )+b \ln \left (b x -1\right )+\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 17, normalized size = 0.94 \begin {gather*} b \log \left (b x - 1\right ) - b \log \relax (x) + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 14, normalized size = 0.78 \begin {gather*} \frac {1}{x}-2\,b\,\mathrm {atanh}\left (2\,b\,x-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} b \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{b} \right )}\right ) + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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