Optimal. Leaf size=11 \[ \log \left (x^2+2\right )-\log (x) \]
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Rubi [A] time = 0.0118638, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {446, 72} \[ \log \left (x^2+2\right )-\log (x) \]
Antiderivative was successfully verified.
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Rule 446
Rule 72
Rubi steps
\begin{align*} \int \frac{-2+x^2}{x \left (2+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-2+x}{x (2+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x}+\frac{2}{2+x}\right ) \, dx,x,x^2\right )\\ &=-\log (x)+\log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0035185, size = 11, normalized size = 1. \[ \log \left (x^2+2\right )-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 12, normalized size = 1.1 \begin{align*} -\ln \left ( x \right ) +\ln \left ({x}^{2}+2 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.924889, size = 18, normalized size = 1.64 \begin{align*} \log \left (x^{2} + 2\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 = 1.83461, size = 31, normalized size = 2.82 \begin{align*} \log \left (x^{2} + 2\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087302, size = 8, normalized size = 0.73 \begin{align*} - \log{\left (x \right )} + \log{\left (x^{2} + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06732, size = 18, normalized size = 1.64 \begin{align*} \log \left (x^{2} + 2\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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