Optimal. Leaf size=21 \[ x \log \left (\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log (x+1) \]
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Rubi [A] time = 0.0078685, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2453, 2448, 263, 31} \[ x \log \left (\sqrt{\frac{1}{x}+1}\right )+\frac{1}{2} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 2453
Rule 2448
Rule 263
Rule 31
Rubi steps
\begin{align*} \int \log \left (\sqrt{\frac{1+x}{x}}\right ) \, dx &=\int \log \left (\sqrt{1+\frac{1}{x}}\right ) \, dx\\ &=x \log \left (\sqrt{1+\frac{1}{x}}\right )+\frac{1}{2} \int \frac{1}{\left (1+\frac{1}{x}\right ) x} \, dx\\ &=x \log \left (\sqrt{1+\frac{1}{x}}\right )+\frac{1}{2} \int \frac{1}{1+x} \, dx\\ &=x \log \left (\sqrt{1+\frac{1}{x}}\right )+\frac{1}{2} \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0032725, size = 19, normalized size = 0.9 \[ \frac{1}{2} \left (\log (x)+(x+1) \log \left (\frac{x+1}{x}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 22, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({x}^{-1} \right ) }{2}}+{\frac{\ln \left ( 1+{x}^{-1} \right ) \left ( 1+{x}^{-1} \right ) x}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00957, size = 24, normalized size = 1.14 \begin{align*} \frac{1}{2} \, x \log \left (\frac{x + 1}{x}\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03842, size = 53, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x \log \left (\frac{x + 1}{x}\right ) + \frac{1}{2} \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.107513, size = 17, normalized size = 0.81 \begin{align*} \frac{x \log{\left (\frac{x + 1}{x} \right )}}{2} + \frac{\log{\left (2 x + 2 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25201, size = 26, normalized size = 1.24 \begin{align*} \frac{1}{2} \, x \log \left (\frac{x + 1}{x}\right ) + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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