Optimal. Leaf size=9 \[ \log \left (1+\log \left (\frac {x}{a}\right )\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2339, 29}
\begin {gather*} \log \left (\log \left (\frac {x}{a}\right )+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+\log \left (\frac {x}{a}\right )\right )} \, dx &=\text {Subst}\left (\int \frac {1}{x} \, dx,x,1+\log \left (\frac {x}{a}\right )\right )\\ &=\log \left (1+\log \left (\frac {x}{a}\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 9, normalized size = 1.00 \begin {gather*} \log \left (1+\log \left (\frac {x}{a}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 10, normalized size = 1.11
method | result | size |
derivativedivides | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
default | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
norman | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
risch | \(\ln \left (1+\ln \left (\frac {x}{a}\right )\right )\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.09, size = 9, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.65, size = 9, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 7, normalized size = 0.78 \begin {gather*} \log {\left (\log {\left (\frac {x}{a} \right )} + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 9, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\frac {x}{a}\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 9, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (\frac {x}{a}\right )+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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