Optimal. Leaf size=15 \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]
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Rubi [A] time = 0.0090262, antiderivative size = 15, normalized size of antiderivative = 1., 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} \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
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 )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\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*}
Mathematica [A] time = 0.0041693, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 14, normalized size = 0.9 \begin{align*} \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 = 2.05713, size = 23, normalized size = 1.53 \begin{align*} -\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 = 1.24878, size = 41, normalized size = 2.73 \begin{align*} -\frac{1}{2} \, \log \left (b x^{2} + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.120866, size = 12, normalized size = 0.8 \begin{align*} \log{\left (x \right )} - \frac{\log{\left (x^{2} + \frac{1}{b} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.17215, size = 24, normalized size = 1.6 \begin{align*} \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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