Optimal. Leaf size=18 \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]
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Rubi [A] time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2486, 31} \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 2486
Rubi steps
\begin {align*} \int \log \left (-\frac {x}{1+x}\right ) \, dx &=x \log \left (-\frac {x}{1+x}\right )-\int \frac {1}{1+x} \, dx\\ &=x \log \left (-\frac {x}{1+x}\right )-\log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x+1}\right )-\log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x + 1}\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 80, normalized size = 4.44 \[ -\frac {\log \left (-\frac {x}{{\left (x + 1\right )} {\left (\frac {x}{x + 1} - 1\right )} {\left (\frac {x}{{\left (x + 1\right )} {\left (\frac {x}{x + 1} - 1\right )}} - 1\right )}}\right )}{\frac {x}{x + 1} - 1} - \log \left (\frac {{\left | x \right |}}{{\left | x + 1 \right |}}\right ) + \log \left ({\left | -\frac {x}{x + 1} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 28, normalized size = 1.56 \[ \ln \left (\frac {1}{x +1}\right )-\left (-1+\frac {1}{x +1}\right ) \left (x +1\right ) \ln \left (-1+\frac {1}{x +1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 18, normalized size = 1.00 \[ x \log \left (-\frac {x}{x + 1}\right ) - \log \left (x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.36, size = 18, normalized size = 1.00 \[ x\,\ln \left (-\frac {x}{x+1}\right )-\ln \left (x+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 14, normalized size = 0.78 \[ x \log {\left (- \frac {x}{x + 1} \right )} - \log {\left (x + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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