Optimal. Leaf size=7 \[ \sinh ^{-1}\left (\frac{\log (x)}{2}\right ) \]
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Rubi [A] time = 0.0291678, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {215} \[ \sinh ^{-1}\left (\frac{\log (x)}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rubi steps
\begin{align*} \int \frac{1}{x \sqrt{4+\log ^2(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{4+x^2}} \, dx,x,\log (x)\right )\\ &=\sinh ^{-1}\left (\frac{\log (x)}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0153227, size = 7, normalized size = 1. \[ \sinh ^{-1}\left (\frac{\log (x)}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 6, normalized size = 0.9 \begin{align*}{\it Arcsinh} \left ({\frac{\ln \left ( x \right ) }{2}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61202, size = 7, normalized size = 1. \begin{align*} \operatorname{arsinh}\left (\frac{1}{2} \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06776, size = 47, normalized size = 6.71 \begin{align*} -\log \left (\sqrt{\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\log{\left (x \right )}^{2} + 4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23523, size = 22, normalized size = 3.14 \begin{align*} -\log \left (\sqrt{\log \left (x\right )^{2} + 4} - \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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