Integrand size = 22, antiderivative size = 24 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {272, 65, 212} \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
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Rule 65
Rule 212
Rule 272
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \sqrt {a^2-x^2}} \, dx,x,\log (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {a^2-x} x} \, dx,x,\log ^2(x)\right ) \\ & = -\text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,\sqrt {a^2-\log ^2(x)}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a^2-\log ^2(x)}}{a}\right )}{a} \]
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Time = 0.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
default | \(-\frac {\ln \left (\frac {2 a^{2}+2 \sqrt {a^{2}}\, \sqrt {a^{2}-\ln \left (x \right )^{2}}}{\ln \left (x \right )}\right )}{\sqrt {a^{2}}}\) | \(39\) |
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none
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=\frac {\log \left (-\frac {a - \sqrt {a^{2} - \log \left (x\right )^{2}}}{\log \left (x\right )}\right )}{a} \]
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\[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=\int \frac {1}{x \sqrt {\left (a - \log {\left (x \right )}\right ) \left (a + \log {\left (x \right )}\right )} \log {\left (x \right )}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=-\frac {\log \left (\frac {2 \, a^{2}}{{\left | \log \left (x\right ) \right |}} + \frac {2 \, \sqrt {a^{2} - \log \left (x\right )^{2}} a}{{\left | \log \left (x\right ) \right |}}\right )}{a} \]
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Timed out. \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=\text {Timed out} \]
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Time = 0.69 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \log (x) \sqrt {a^2-\log ^2(x)}} \, dx=-\frac {\mathrm {atanh}\left (\frac {\sqrt {a^2-{\ln \left (x\right )}^2}}{a}\right )}{a} \]
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