Optimal. Leaf size=29 \[ \frac{1}{24} (4 \log (x)-1)^{3/2}+\frac{1}{8} \sqrt{4 \log (x)-1} \]
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Rubi [A] time = 0.0505343, antiderivative size = 29, 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 = {2365, 43} \[ \frac{1}{24} (4 \log (x)-1)^{3/2}+\frac{1}{8} \sqrt{4 \log (x)-1} \]
Antiderivative was successfully verified.
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Rule 2365
Rule 43
Rubi steps
\begin{align*} \int \frac{\log (x)}{x \sqrt{-1+4 \log (x)}} \, dx &=\operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+4 x}} \, dx,x,\log (x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{4 \sqrt{-1+4 x}}+\frac{1}{4} \sqrt{-1+4 x}\right ) \, dx,x,\log (x)\right )\\ &=\frac{1}{8} \sqrt{-1+4 \log (x)}+\frac{1}{24} (-1+4 \log (x))^{3/2}\\ \end{align*}
Mathematica [A] time = 0.021203, size = 20, normalized size = 0.69 \[ \frac{1}{12} (2 \log (x)+1) \sqrt{4 \log (x)-1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 22, normalized size = 0.8 \begin{align*}{\frac{1}{24} \left ( -1+4\,\ln \left ( x \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{1}{8}\sqrt{-1+4\,\ln \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01078, size = 28, normalized size = 0.97 \begin{align*} \frac{1}{24} \,{\left (4 \, \log \left (x\right ) - 1\right )}^{\frac{3}{2}} + \frac{1}{8} \, \sqrt{4 \, \log \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01502, size = 54, normalized size = 1.86 \begin{align*} \frac{1}{12} \, \sqrt{4 \, \log \left (x\right ) - 1}{\left (2 \, \log \left (x\right ) + 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.35269, size = 22, normalized size = 0.76 \begin{align*} \frac{\left (4 \log{\left (x \right )} - 1\right )^{\frac{3}{2}}}{24} + \frac{\sqrt{4 \log{\left (x \right )} - 1}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19953, size = 28, normalized size = 0.97 \begin{align*} \frac{1}{24} \,{\left (4 \, \log \left (x\right ) - 1\right )}^{\frac{3}{2}} + \frac{1}{8} \, \sqrt{4 \, \log \left (x\right ) - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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