Optimal. Leaf size=18 \[ \tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {223, 209}
\begin {gather*} \text {ArcTan}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2-\log ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\sqrt {a^2-x^2}} \, dx,x,\log (x)\right )\\ &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {\log (x)}{\sqrt {a^2-\log ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\arctan \left (\frac {\ln \left (x \right )}{\sqrt {a^{2}-\ln \left (x \right )^{2}}}\right )\) | \(17\) |
default | \(\arctan \left (\frac {\ln \left (x \right )}{\sqrt {a^{2}-\ln \left (x \right )^{2}}}\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.43, size = 7, normalized size = 0.39 \begin {gather*} \arcsin \left (\frac {\log \left (x\right )}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.63, size = 25, normalized size = 1.39 \begin {gather*} -2 \, \arctan \left (-\frac {a - \sqrt {a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {\left (a - \log {\left (x \right )}\right ) \left (a + \log {\left (x \right )}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.71, size = 10, normalized size = 0.56 \begin {gather*} \arcsin \left (\frac {\log \left (x\right )}{a}\right ) \mathrm {sgn}\left (a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.61, size = 16, normalized size = 0.89 \begin {gather*} \mathrm {atan}\left (\frac {\ln \left (x\right )}{\sqrt {a^2-{\ln \left (x\right )}^2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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