Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\text {arctanh}\left (\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {223, 212} \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\text {arctanh}\left (\frac {\log (x)}{\sqrt {\log ^2(x)-a^2}}\right ) \]
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Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \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 ) \\ & = \text {arctanh}\left (\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(50\) vs. \(2(18)=36\).
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=-\frac {1}{2} \log \left (1-\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right )+\frac {1}{2} \log \left (1+\frac {\log (x)}{\sqrt {-a^2+\log ^2(x)}}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\ln \left (\ln \left (x \right )+\sqrt {-a^{2}+\ln \left (x \right )^{2}}\right )\) | \(17\) |
default | \(\ln \left (\ln \left (x \right )+\sqrt {-a^{2}+\ln \left (x \right )^{2}}\right )\) | \(17\) |
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none
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=-\log \left (\sqrt {-a^{2} + \log \left (x\right )^{2}} - \log \left (x\right )\right ) \]
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\[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\int \frac {1}{x \sqrt {- \left (a - \log {\left (x \right )}\right ) \left (a + \log {\left (x \right )}\right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\log \left (2 \, \sqrt {-a^{2} + \log \left (x\right )^{2}} + 2 \, \log \left (x\right )\right ) \]
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Timed out. \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\text {Timed out} \]
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Time = 0.45 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {-a^2+\log ^2(x)}} \, dx=\ln \left (\ln \left (x\right )+\sqrt {{\ln \left (x\right )}^2-a^2}\right ) \]
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