Optimal. Leaf size=8 \[ \frac {x}{\log (x)+1} \]
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Rubi [A] time = 0.04, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2360, 2297, 2299, 2178} \[ \frac {x}{\log (x)+1} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2297
Rule 2299
Rule 2360
Rubi steps
\begin {align*} \int \frac {\log (x)}{(1+\log (x))^2} \, dx &=\int \left (-\frac {1}{(1+\log (x))^2}+\frac {1}{1+\log (x)}\right ) \, dx\\ &=-\int \frac {1}{(1+\log (x))^2} \, dx+\int \frac {1}{1+\log (x)} \, dx\\ &=\frac {x}{1+\log (x)}-\int \frac {1}{1+\log (x)} \, dx+\operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,\log (x)\right )\\ &=\frac {\text {Ei}(1+\log (x))}{e}+\frac {x}{1+\log (x)}-\operatorname {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,\log (x)\right )\\ &=\frac {x}{1+\log (x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 8, normalized size = 1.00 \[ \frac {x}{\log (x)+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 8, normalized size = 1.00 \[ \frac {x}{\log \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 8, normalized size = 1.00 \[ \frac {x}{\log \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 9, normalized size = 1.12 \[ \frac {x}{\ln \relax (x )+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 8, normalized size = 1.00 \[ \frac {x}{\log \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 8, normalized size = 1.00 \[ \frac {x}{\ln \relax (x)+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 5, normalized size = 0.62 \[ \frac {x}{\log {\relax (x )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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