Optimal. Leaf size=13 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6686} \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
Antiderivative was successfully verified.
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Rule 6686
Rubi steps
\begin {align*} \int \left (\frac {1}{x}+\frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}}\right ) \, dx &=\log (x)+\int \frac {1+\frac {1}{x}}{(x+\log (x))^{3/2}} \, dx\\ &=\log (x)-\frac {2}{\sqrt {x+\log (x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.00 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 24, normalized size = 1.85 \[ \frac {x \log \relax (x) + \log \relax (x)^{2} - 2 \, \sqrt {x + \log \relax (x)}}{x + \log \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 12, normalized size = 0.92 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 12, normalized size = 0.92 \[ \ln \relax (x )-\frac {2}{\sqrt {x +\ln \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 11, normalized size = 0.85 \[ -\frac {2}{\sqrt {x + \log \relax (x)}} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 11, normalized size = 0.85 \[ \ln \relax (x)-\frac {2}{\sqrt {x+\ln \relax (x)}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.49, size = 12, normalized size = 0.92 \[ \log {\relax (x )} - \frac {2}{\sqrt {x + \log {\relax (x )}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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