Optimal. Leaf size=11 \[ \frac {\log (a+b \log (x))}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2339, 29}
\begin {gather*} \frac {\log (a+b \log (x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rubi steps
\begin {align*} \int \frac {1}{x (a+b \log (x))} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (x)\right )}{b}\\ &=\frac {\log (a+b \log (x))}{b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log (a+b \log (x))}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 12, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \ln \left (x \right )\right )}{b}\) | \(12\) |
default | \(\frac {\ln \left (a +b \ln \left (x \right )\right )}{b}\) | \(12\) |
norman | \(\frac {\ln \left (a +b \ln \left (x \right )\right )}{b}\) | \(12\) |
risch | \(\frac {\ln \left (\ln \left (x \right )+\frac {a}{b}\right )}{b}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.18, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (b \log \left (x\right ) + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.78, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (b \log \left (x\right ) + a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 8, normalized size = 0.73 \begin {gather*} \frac {\log {\left (\frac {a}{b} + \log {\left (x \right )} \right )}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (11) = 22\).
time = 0.81, size = 30, normalized size = 2.73 \begin {gather*} \frac {\log \left (\frac {1}{4} \, \pi ^{2} b^{2} {\left (\mathrm {sgn}\left (x\right ) - 1\right )}^{2} + {\left (b \log \left ({\left | x \right |}\right ) + a\right )}^{2}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 11, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,\ln \left (x\right )\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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