Optimal. Leaf size=27 \[ -\left ((1-x) \log \left (-\frac {1-x}{1+x}\right )\right )-2 \log (1+x) \]
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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 = {2535, 31}
\begin {gather*} -\left ((1-x) \log \left (-\frac {1-x}{x+1}\right )\right )-2 \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2535
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 \begin {gather*} (-1+x) \log \left (\frac {-1+x}{1+x}\right )-2 \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 35, normalized size = 1.30
method | result | size |
risch | \(x \ln \left (\frac {-1+x}{1+x}\right )-\ln \left (x^{2}-1\right )\) | \(22\) |
derivativedivides | \(2 \ln \left (-\frac {2}{1+x}\right )+\ln \left (1-\frac {2}{1+x}\right ) \left (1-\frac {2}{1+x}\right ) \left (1+x \right )\) | \(35\) |
default | \(2 \ln \left (-\frac {2}{1+x}\right )+\ln \left (1-\frac {2}{1+x}\right ) \left (1-\frac {2}{1+x}\right ) \left (1+x \right )\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 25, normalized size = 0.93 \begin {gather*} x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x + 1\right ) - \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 21, normalized size = 0.78 \begin {gather*} x \log \left (\frac {x - 1}{x + 1}\right ) - \log \left (x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 15, normalized size = 0.56 \begin {gather*} x \log {\left (\frac {x - 1}{x + 1} \right )} - \log {\left (x^{2} - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 103 vs.
\(2 (21) = 42\).
time = 5.91, size = 103, normalized size = 3.81 \begin {gather*} -\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 ) \end {gather*}
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 \begin {gather*} x\,\ln \left (\frac {x-1}{x+1}\right )-\ln \left (x^2-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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