Optimal. Leaf size=3 \[ \tan ^{-1}(\log (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {203} \[ \tan ^{-1}(\log (x)) \]
Antiderivative was successfully verified.
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Rule 203
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\log (x)\right )\\ &=\tan ^{-1}(\log (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 3, normalized size = 1.00 \[ \tan ^{-1}(\log (x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.77, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 4, normalized size = 1.33
method | result | size |
derivativedivides | \(\arctan \left (\ln \relax (x )\right )\) | \(4\) |
default | \(\arctan \left (\ln \relax (x )\right )\) | \(4\) |
risch | \(\frac {i \ln \left (\ln \relax (x )+i\right )}{2}-\frac {i \ln \left (\ln \relax (x )-i\right )}{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 3, normalized size = 1.00 \[ \arctan \left (\log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 3, normalized size = 1.00 \[ \mathrm {atan}\left (\ln \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.14, size = 15, normalized size = 5.00 \[ \operatorname {RootSum} {\left (4 z^{2} + 1, \left (i \mapsto i \log {\left (2 i + \log {\relax (x )} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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