Optimal. Leaf size=18 \[ x \log \left (-\frac{x}{x+1}\right )-\log (x+1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0035066, antiderivative size = 18, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 2486
Rule 31
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*}
Mathematica [A] time = 0.0016447, size = 18, normalized size = 1. \[ x \log \left (-\frac{x}{x+1}\right )-\log (x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 28, normalized size = 1.6 \begin{align*} \ln \left ( \left ( 1+x \right ) ^{-1} \right ) -\ln \left ( -1+ \left ( 1+x \right ) ^{-1} \right ) \left ( -1+ \left ( 1+x \right ) ^{-1} \right ) \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.07712, size = 24, normalized size = 1.33 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.94007, size = 43, normalized size = 2.39 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.105937, size = 14, normalized size = 0.78 \begin{align*} x \log{\left (- \frac{x}{x + 1} \right )} - \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.30919, size = 26, normalized size = 1.44 \begin{align*} x \log \left (-\frac{x}{x + 1}\right ) - \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]