Optimal. Leaf size=27 \[ -\left ((1-x) \log \left (-\frac {1-x}{x+1}\right )\right )-2 \log (x+1) \]
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Rubi [A] time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 31
Rule 2486
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*}
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Mathematica [A] time = 0.00, 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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fricas [A] time = 0.42, size = 21, normalized size = 0.78 \[ x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 103, normalized size = 3.81 \[ -\frac {2 \, \log \left (\frac {\frac {\frac {x - 1}{x + 1} + 1}{\frac {x - 1}{x + 1} - 1} + 1}{\frac {\frac {x - 1}{x + 1} + 1}{\frac {x - 1}{x + 1} - 1} - 1}\right )}{\frac {x - 1}{x + 1} - 1} - 2 \, \log \left (\frac {{\left | x - 1 \right |}}{{\left | x + 1 \right |}}\right ) + 2 \, \log \left ({\left | \frac {x - 1}{x + 1} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 35, normalized size = 1.30 \[ 2 \ln \left (-\frac {2}{x +1}\right )+\left (1-\frac {2}{x +1}\right ) \left (x +1\right ) \ln \left (1-\frac {2}{x +1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 25, normalized size = 0.93 \[ x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x + 1\right ) - \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.78 \[ x\,\ln \left (\frac {x-1}{x+1}\right )-\ln \left (x^2-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 15, normalized size = 0.56 \[ x \log {\left (\frac {x - 1}{x + 1} \right )} - \log {\left (x^{2} - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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