Optimal. Leaf size=36 \[ -\frac{x^2}{6}+\frac{1}{3} x^3 \log \left (\frac{x-1}{x}\right )-\frac{x}{3}-\frac{1}{3} \log (x-1) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0255435, antiderivative size = 38, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2461, 2455, 263, 43} \[ -\frac{x^2}{6}+\frac{1}{3} x^3 \log \left (1-\frac{1}{x}\right )-\frac{x}{3}-\frac{1}{3} \log (1-x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2461
Rule 2455
Rule 263
Rule 43
Rubi steps
\begin{align*} \int x^2 \log \left (\frac{-1+x}{x}\right ) \, dx &=\int x^2 \log \left (1-\frac{1}{x}\right ) \, dx\\ &=\frac{1}{3} x^3 \log \left (1-\frac{1}{x}\right )-\frac{1}{3} \int \frac{x}{1-\frac{1}{x}} \, dx\\ &=\frac{1}{3} x^3 \log \left (1-\frac{1}{x}\right )-\frac{1}{3} \int \frac{x^2}{-1+x} \, dx\\ &=\frac{1}{3} x^3 \log \left (1-\frac{1}{x}\right )-\frac{1}{3} \int \left (1+\frac{1}{-1+x}+x\right ) \, dx\\ &=-\frac{x}{3}-\frac{x^2}{6}+\frac{1}{3} x^3 \log \left (1-\frac{1}{x}\right )-\frac{1}{3} \log (1-x)\\ \end{align*}
Mathematica [A] time = 0.0056543, size = 38, normalized size = 1.06 \[ -\frac{x^2}{6}+\frac{1}{3} x^3 \log \left (\frac{x-1}{x}\right )-\frac{x}{3}-\frac{1}{3} \log (1-x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.012, size = 53, normalized size = 1.5 \begin{align*} -{\frac{{x}^{2}}{6}}+{\frac{1}{3}\ln \left ( -{x}^{-1} \right ) }-{\frac{x}{3}}+{\frac{{x}^{3}}{3}\ln \left ( 1-{x}^{-1} \right ) \left ( 1-{x}^{-1} \right ) \left ( \left ( 1-{x}^{-1} \right ) ^{2}+3\,{x}^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.940232, size = 38, normalized size = 1.06 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\frac{x - 1}{x}\right ) - \frac{1}{6} \, x^{2} - \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.88659, size = 80, normalized size = 2.22 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\frac{x - 1}{x}\right ) - \frac{1}{6} \, x^{2} - \frac{1}{3} \, x - \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.117856, size = 26, normalized size = 0.72 \begin{align*} \frac{x^{3} \log{\left (\frac{x - 1}{x} \right )}}{3} - \frac{x^{2}}{6} - \frac{x}{3} - \frac{\log{\left (x - 1 \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.05709, size = 39, normalized size = 1.08 \begin{align*} \frac{1}{3} \, x^{3} \log \left (\frac{x - 1}{x}\right ) - \frac{1}{6} \, x^{2} - \frac{1}{3} \, x - \frac{1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]