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 = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 31
Rule 263
Rule 2448
Rule 2453
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*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 0.90 \[ \frac {1}{2} \left (\log (x)+(x+1) \log \left (\frac {x+1}{x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 0.86 \[ \frac {1}{2} \, x \log \left (\frac {x + 1}{x}\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 47, normalized size = 2.24 \[ \frac {\log \left (\frac {x + 1}{x}\right )}{2 \, {\left (\frac {x + 1}{x} - 1\right )}} + \frac {1}{2} \, \log \left (\frac {{\left | x + 1 \right |}}{{\left | x \right |}}\right ) - \frac {1}{2} \, \log \left ({\left | \frac {x + 1}{x} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 22, normalized size = 1.05 \[ \frac {\left (\frac {1}{x}+1\right ) x \ln \left (\frac {1}{x}+1\right )}{2}-\frac {\ln \left (\frac {1}{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 18, normalized size = 0.86 \[ \frac {1}{2} \, x \log \left (\frac {x + 1}{x}\right ) + \frac {1}{2} \, \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 18, normalized size = 0.86 \[ \frac {\ln \left (x+1\right )}{2}+\frac {x\,\ln \left (\frac {x+1}{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 17, normalized size = 0.81 \[ \frac {x \log {\left (\frac {x + 1}{x} \right )}}{2} + \frac {\log {\left (2 x + 2 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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