Optimal. Leaf size=22 \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]
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Rubi [A] time = 0.0561296, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2365, 50, 63, 207} \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]
Antiderivative was successfully verified.
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Rule 2365
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\sqrt{1+\log (x)}}{x \log (x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{x} \, dx,x,\log (x)\right )\\ &=2 \sqrt{1+\log (x)}+\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,\log (x)\right )\\ &=2 \sqrt{1+\log (x)}+2 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+\log (x)}\right )\\ &=-2 \tanh ^{-1}\left (\sqrt{1+\log (x)}\right )+2 \sqrt{1+\log (x)}\\ \end{align*}
Mathematica [A] time = 0.021054, size = 22, normalized size = 1. \[ 2 \sqrt{\log (x)+1}-2 \tanh ^{-1}\left (\sqrt{\log (x)+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 30, normalized size = 1.4 \begin{align*} 2\,\sqrt{1+\ln \left ( x \right ) }+\ln \left ( -1+\sqrt{1+\ln \left ( x \right ) } \right ) -\ln \left ( 1+\sqrt{1+\ln \left ( x \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949866, size = 39, normalized size = 1.77 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97212, size = 103, normalized size = 4.68 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79135, size = 32, normalized size = 1.45 \begin{align*} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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