Optimal. Leaf size=38 \[ -\frac{\log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x)}{a^2}+\frac{1}{2 a \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0382369, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 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 28
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac{1}{x \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 b^2 x}-\frac{1}{a b (a+b x)^2}-\frac{1}{a^2 b (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 = 0.0126737, size = 33, normalized size = 0.87 \[ \frac{\frac{a}{a+b x^2}-\log \left (a+b x^2\right )+2 \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{2}}}-{\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 = 0.989865, size = 50, normalized size = 1.32 \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 = 1.688, size = 108, normalized size = 2.84 \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 = 0.440688, size = 34, normalized size = 0.89 \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 = 1.14674, size = 63, normalized size = 1.66 \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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