Optimal. Leaf size=30 \[ \frac{1}{2 \left (1-b x^2\right )}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]
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Rubi [A] time = 0.0194258, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 44} \[ \frac{1}{2 \left (1-b x^2\right )}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (-1+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (-1+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{x}+\frac{b}{(-1+b x)^2}-\frac{b}{-1+b x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2 \left (1-b x^2\right )}+\log (x)-\frac{1}{2} \log \left (1-b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0121764, size = 26, normalized size = 0.87 \[ \frac{1}{2-2 b x^2}-\frac{1}{2} \log \left (1-b x^2\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 25, normalized size = 0.8 \begin{align*} \ln \left ( x \right ) -{\frac{\ln \left ( b{x}^{2}-1 \right ) }{2}}-{\frac{1}{2\,b{x}^{2}-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.48238, size = 38, normalized size = 1.27 \begin{align*} -\frac{1}{2 \,{\left (b x^{2} - 1\right )}} - \frac{1}{2} \, \log \left (b x^{2} - 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40382, size = 100, normalized size = 3.33 \begin{align*} -\frac{{\left (b x^{2} - 1\right )} \log \left (b x^{2} - 1\right ) - 2 \,{\left (b x^{2} - 1\right )} \log \left (x\right ) + 1}{2 \,{\left (b x^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.354297, size = 22, normalized size = 0.73 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} - \frac{1}{b} \right )}}{2} - \frac{1}{2 b x^{2} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.77783, size = 49, normalized size = 1.63 \begin{align*} \frac{b x^{2} - 2}{2 \,{\left (b x^{2} - 1\right )}} + \frac{1}{2} \, \log \left (x^{2}\right ) - \frac{1}{2} \, \log \left ({\left | b x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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