Optimal. Leaf size=23 \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]
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Rubi [A] time = 0.0122526, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]
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 (a-b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a-b x)} \, dx,x,x^2\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a-b x} \, dx,x,x^2\right )}{2 a}\\ &=\frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.006051, size = 23, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{\log \left (a-b x^2\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 23, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.50863, size = 34, normalized size = 1.48 \begin{align*} -\frac{\log \left (b x^{2} - a\right )}{2 \, a} + \frac{\log \left (x^{2}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26249, size = 49, normalized size = 2.13 \begin{align*} -\frac{\log \left (b x^{2} - a\right ) - 2 \, \log \left (x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.192758, size = 15, normalized size = 0.65 \begin{align*} \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.8525, size = 35, normalized size = 1.52 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a} - \frac{\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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