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 = 0.05, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 43
Rule 2365
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.64, size = 16, normalized size = 0.55 \[ \frac {1}{12} \, \sqrt {4 \, \log \relax (x) - 1} {\left (2 \, \log \relax (x) + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 21, normalized size = 0.72 \[ \frac {1}{24} \, {\left (4 \, \log \relax (x) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 22, normalized size = 0.76 \[ \frac {\left (4 \ln \relax (x )-1\right )^{\frac {3}{2}}}{24}+\frac {\sqrt {4 \ln \relax (x )-1}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.75, size = 21, normalized size = 0.72 \[ \frac {1}{24} \, {\left (4 \, \log \relax (x) - 1\right )}^{\frac {3}{2}} + \frac {1}{8} \, \sqrt {4 \, \log \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 15, normalized size = 0.52 \[ \sqrt {4\,\ln \relax (x)-1}\,\left (\frac {\ln \relax (x)}{6}+\frac {1}{12}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.43, size = 22, normalized size = 0.76 \[ \frac {\left (4 \log {\relax (x )} - 1\right )^{\frac {3}{2}}}{24} + \frac {\sqrt {4 \log {\relax (x )} - 1}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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