Optimal. Leaf size=3 \[ \tan ^{-1}(\log (x)) \]
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Rubi [A]
time = 0.01, 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 = {209}
\begin {gather*} \text {ArcTan}(\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx &=\text {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 \begin {gather*} \tan ^{-1}(\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 4, normalized size = 1.33
method | result | size |
derivativedivides | \(\arctan \left (\ln \left (x \right )\right )\) | \(4\) |
default | \(\arctan \left (\ln \left (x \right )\right )\) | \(4\) |
risch | \(\frac {i \ln \left (\ln \left (x \right )+i\right )}{2}-\frac {i \ln \left (\ln \left (x \right )-i\right )}{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.09, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.90, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs.
\(2 (3) = 6\).
time = 0.05, size = 15, normalized size = 5.00 \begin {gather*} \operatorname {RootSum} {\left (4 z^{2} + 1, \left ( i \mapsto i \log {\left (2 i + \log {\left (x \right )} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\log \left (x\right )\right ) \end {gather*}
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 \begin {gather*} \mathrm {atan}\left (\ln \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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