Optimal. Leaf size=40 \[ -\frac{\log \left (a-b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a-b x^2\right )} \]
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Rubi [A] time = 0.0279616, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 44} \[ -\frac{\log \left (a-b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a-b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a-b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (a-b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}+\frac{b}{a (a-b x)^2}+\frac{b}{a^2 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2 a \left (a-b x^2\right )}+\frac{\log (x)}{a^2}-\frac{\log \left (a-b x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0160346, size = 35, normalized size = 0.88 \[ \frac{\frac{a}{a-b x^2}-\log \left (a-b x^2\right )+2 \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 39, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\frac{1}{2\,a \left ( b{x}^{2}-a \right ) }}-{\frac{\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20726, size = 55, normalized size = 1.38 \begin{align*} -\frac{1}{2 \,{\left (a b x^{2} - a^{2}\right )}} - \frac{\log \left (b x^{2} - a\right )}{2 \, a^{2}} + \frac{\log \left (x^{2}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24117, size = 108, normalized size = 2.7 \begin{align*} -\frac{{\left (b x^{2} - a\right )} \log \left (b x^{2} - a\right ) - 2 \,{\left (b x^{2} - a\right )} \log \left (x\right ) + a}{2 \,{\left (a^{2} b x^{2} - a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.447608, size = 34, normalized size = 0.85 \begin{align*} - \frac{1}{- 2 a^{2} + 2 a b x^{2}} + \frac{\log{\left (x \right )}}{a^{2}} - \frac{\log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.76258, size = 69, normalized size = 1.72 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a^{2}} - \frac{\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{2}} + \frac{b x^{2} - 2 \, a}{2 \,{\left (b x^{2} - a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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