Optimal. Leaf size=27 \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]
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Rubi [A] time = 0.017959, antiderivative size = 27, 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} \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (-1+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (-1+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x^2}-\frac{b}{x}+\frac{b^2}{-1+b x}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2 x^2}-b \log (x)+\frac{1}{2} b \log \left (1-b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.004213, size = 27, normalized size = 1. \[ \frac{1}{2} b \log \left (1-b x^2\right )-b \log (x)+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 23, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{2}}}-b\ln \left ( x \right ) +{\frac{b\ln \left ( b{x}^{2}-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.2964, size = 32, normalized size = 1.19 \begin{align*} \frac{1}{2} \, b \log \left (b x^{2} - 1\right ) - \frac{1}{2} \, b \log \left (x^{2}\right ) + \frac{1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24557, size = 72, normalized size = 2.67 \begin{align*} \frac{b x^{2} \log \left (b x^{2} - 1\right ) - 2 \, b x^{2} \log \left (x\right ) + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.193962, size = 22, normalized size = 0.81 \begin{align*} - b \log{\left (x \right )} + \frac{b \log{\left (x^{2} - \frac{1}{b} \right )}}{2} + \frac{1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85536, size = 43, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, b \log \left (x^{2}\right ) + \frac{1}{2} \, b \log \left ({\left | b x^{2} - 1 \right |}\right ) + \frac{b x^{2} + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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