Optimal. Leaf size=28 \[ \frac{1}{2 \left (b x^2+1\right )}-\frac{1}{2} \log \left (b x^2+1\right )+\log (x) \]
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Rubi [A] time = 0.0199275, antiderivative size = 28, 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} \[ \frac{1}{2 \left (b x^2+1\right )}-\frac{1}{2} \log \left (b x^2+1\right )+\log (x) \]
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 = 0.0125481, size = 25, normalized size = 0.89 \[ \frac{1}{2 b x^2+2}-\frac{1}{2} \log \left (b x^2+1\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 25, normalized size = 0.9 \begin{align*}{\frac{1}{2\,b{x}^{2}+2}}+\ln \left ( x \right ) -{\frac{\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 = 2.5868, size = 38, normalized size = 1.36 \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 = 1.24831, size = 100, normalized size = 3.57 \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 = 0.344172, size = 22, normalized size = 0.79 \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 = 2.26561, size = 49, normalized size = 1.75 \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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