Optimal. Leaf size=18 \[ \frac {1}{2} \log \left (1-b x^2\right )-\log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac {1}{2} \log \left (1-b x^2\right )-\log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (-1+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (-1+b x)} \, dx,x,x^2\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )\right )+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{-1+b x} \, dx,x,x^2\right )\\ &=-\log (x)+\frac {1}{2} \log \left (1-b x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ \frac {1}{2} \log \left (1-b x^2\right )-\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 15, normalized size = 0.83 \[ \frac {1}{2} \, \log \left (b x^{2} - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 18, normalized size = 1.00 \[ -\frac {1}{2} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | b x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.89 \[ -\ln \relax (x )+\frac {\ln \left (b \,x^{2}-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 17, normalized size = 0.94 \[ \frac {1}{2} \, \log \left (b x^{2} - 1\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 16, normalized size = 0.89 \[ \frac {\ln \left (\frac {3}{2}-\frac {3\,b\,x^2}{2}\right )}{2}-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 12, normalized size = 0.67 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} - \frac {1}{b} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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