Optimal. Leaf size=18 \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}\right ) \]
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Rubi [A] time = 0.04, 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 = {217, 206} \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-a^2+x^2}} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.04, size = 50, normalized size = 2.78 \[ \frac {1}{2} \log \left (\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}+1\right )-\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.00, size = 20, normalized size = 1.11 \[ -\log \left (\sqrt {-a^{2} + \log \relax (x)^{2}} - \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\ln \left (\ln \relax (x )+\sqrt {-a^{2}+\ln \relax (x )^{2}}\right )\) | \(17\) |
default | \(\ln \left (\ln \relax (x )+\sqrt {-a^{2}+\ln \relax (x )^{2}}\right )\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 20, normalized size = 1.11 \[ \log \left (2 \, \sqrt {-a^{2} + \log \relax (x)^{2}} + 2 \, \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 16, normalized size = 0.89 \[ \ln \left (\ln \relax (x)+\sqrt {{\ln \relax (x)}^2-a^2}\right ) \]
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 \[ \int \frac {1}{x \sqrt {- \left (a - \log {\relax (x )}\right ) \left (a + \log {\relax (x )}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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