Optimal. Leaf size=27 \[ -(1-x) \log \left (-\frac{1-x}{x+1}\right )-2 \log (x+1) \]
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Rubi [A] time = 0.0049812, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2486, 31} \[ -(1-x) \log \left (-\frac{1-x}{x+1}\right )-2 \log (x+1) \]
Antiderivative was successfully verified.
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Rule 2486
Rule 31
Rubi steps
\begin{align*} \int \log \left (\frac{-1+x}{1+x}\right ) \, dx &=-(1-x) \log \left (-\frac{1-x}{1+x}\right )-2 \int \frac{1}{1+x} \, dx\\ &=-(1-x) \log \left (-\frac{1-x}{1+x}\right )-2 \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0027194, size = 21, normalized size = 0.78 \[ (x-1) \log \left (\frac{x-1}{x+1}\right )-2 \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 35, normalized size = 1.3 \begin{align*} 2\,\ln \left ( -2\, \left ( 1+x \right ) ^{-1} \right ) +\ln \left ( 1-2\, \left ( 1+x \right ) ^{-1} \right ) \left ( 1-2\, \left ( 1+x \right ) ^{-1} \right ) \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00724, size = 34, normalized size = 1.26 \begin{align*} x \log \left (\frac{x - 1}{x + 1}\right ) - \log \left (x + 1\right ) - \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10258, size = 53, normalized size = 1.96 \begin{align*} x \log \left (\frac{x - 1}{x + 1}\right ) - \log \left (x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.119236, size = 15, normalized size = 0.56 \begin{align*} x \log{\left (\frac{x - 1}{x + 1} \right )} - \log{\left (x^{2} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32349, size = 30, normalized size = 1.11 \begin{align*} x \log \left (\frac{x - 1}{x + 1}\right ) - \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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