Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\text {arctanh}\left (\frac {\log (x)}{\sqrt {-3+\log ^2(x)}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {223, 212} \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\text {arctanh}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-3}}\right ) \]
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Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {-3+x^2}} \, dx,x,\log (x)\right ) \\ & = \text {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*}
Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(14)=28\).
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 3.00 \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=-\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {-3+\log ^2(x)}}\right )+\frac {1}{2} \log \left (1+\frac {\log (x)}{\sqrt {-3+\log ^2(x)}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\ln \left (\ln \left (x \right )+\sqrt {-3+\ln \left (x \right )^{2}}\right )\) | \(13\) |
default | \(\ln \left (\ln \left (x \right )+\sqrt {-3+\ln \left (x \right )^{2}}\right )\) | \(13\) |
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none
Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=-\log \left (\sqrt {\log \left (x\right )^{2} - 3} - \log \left (x\right )\right ) \]
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\[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\int \frac {1}{x \sqrt {\log {\left (x \right )}^{2} - 3}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\log \left (2 \, \sqrt {\log \left (x\right )^{2} - 3} + 2 \, \log \left (x\right )\right ) \]
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Timed out. \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\text {Timed out} \]
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Time = 1.63 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x \sqrt {-3+\log ^2(x)}} \, dx=\ln \left (\ln \left (x\right )+\sqrt {{\ln \left (x\right )}^2-3}\right ) \]
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