Optimal. Leaf size=8 \[ \frac {x}{1+\log (x)} \]
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Rubi [A]
time = 0.03, 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 = {2407, 2334,
2336, 2209} \begin {gather*} \frac {x}{\log (x)+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2334
Rule 2336
Rule 2407
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+\text {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)}-\text {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 = 0.01, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{1+\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.73, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{1+\text {Log}\left [x\right ]} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 9, normalized size = 1.12
method | result | size |
default | \(\frac {x}{1+\ln \left (x \right )}\) | \(9\) |
norman | \(\frac {x}{1+\ln \left (x \right )}\) | \(9\) |
risch | \(\frac {x}{1+\ln \left (x \right )}\) | \(9\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 8, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (x\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 5, normalized size = 0.62 \begin {gather*} \frac {x}{\log {\left (x \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 7, normalized size = 0.88 \begin {gather*} \frac {x}{\ln x+1} \end {gather*}
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 \begin {gather*} \frac {x}{\ln \left (x\right )+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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