Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-3}}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {217, 206} \[ \tanh ^{-1}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-3+x^2}} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\log (x)}{\sqrt {-3+\log ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac {\log (x)}{\sqrt {-3+\log ^2(x)}}\right )\\ \end {align*}
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Mathematica [B] time = 0.02, size = 42, normalized size = 3.00 \[ \frac {1}{2} \log \left (\frac {\log (x)}{\sqrt {\log ^2(x)-3}}+1\right )-\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {\log ^2(x)-3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 16, normalized size = 1.14 \[ -\log \left (\sqrt {\log \relax (x)^{2} - 3} - \log \relax (x)\right ) \]
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 = 0.07, size = 13, normalized size = 0.93 \[ \ln \left (\ln \relax (x )+\sqrt {\ln \relax (x )^{2}-3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 16, normalized size = 1.14 \[ \log \left (2 \, \sqrt {\log \relax (x)^{2} - 3} + 2 \, \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 12, normalized size = 0.86 \[ \ln \left (\ln \relax (x)+\sqrt {{\ln \relax (x)}^2-3}\right ) \]
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 \[ \int \frac {1}{x \sqrt {\log {\relax (x )}^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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