Integrand size = 16, antiderivative size = 22 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=-2 \text {arctanh}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)} \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2412, 52, 65, 213} \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \sqrt {\log (x)+1}-2 \text {arctanh}\left (\sqrt {\log (x)+1}\right ) \]
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Rule 52
Rule 65
Rule 213
Rule 2412
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,\log (x)\right ) \\ & = 2 \sqrt {1+\log (x)}+\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\log (x)\right ) \\ & = 2 \sqrt {1+\log (x)}+2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\log (x)}\right ) \\ & = -2 \text {arctanh}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=-2 \text {arctanh}\left (\sqrt {1+\log (x)}\right )+2 \sqrt {1+\log (x)} \]
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Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
derivativedivides | \(2 \sqrt {1+\ln \left (x \right )}+\ln \left (\sqrt {1+\ln \left (x \right )}-1\right )-\ln \left (\sqrt {1+\ln \left (x \right )}+1\right )\) | \(30\) |
default | \(2 \sqrt {1+\ln \left (x \right )}+\ln \left (\sqrt {1+\ln \left (x \right )}-1\right )-\ln \left (\sqrt {1+\ln \left (x \right )}+1\right )\) | \(30\) |
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none
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \, \sqrt {\log \left (x\right ) + 1} - \log \left (\sqrt {\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt {\log \left (x\right ) + 1} - 1\right ) \]
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Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \sqrt {\log {\left (x \right )} + 1} + \log {\left (\sqrt {\log {\left (x \right )} + 1} - 1 \right )} - \log {\left (\sqrt {\log {\left (x \right )} + 1} + 1 \right )} \]
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Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2 \, \sqrt {\log \left (x\right ) + 1} - \log \left (\sqrt {\log \left (x\right ) + 1} + 1\right ) + \log \left (\sqrt {\log \left (x\right ) + 1} - 1\right ) \]
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Timed out. \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {1+\log (x)}}{x \log (x)} \, dx=2\,\sqrt {\ln \left (x\right )+1}-2\,\mathrm {atanh}\left (\sqrt {\ln \left (x\right )+1}\right ) \]
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