Optimal. Leaf size=24 \[ -3 x+x \log (x)+(x+1) \log (x+1)+(x+2) \log (x+2) \]
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Rubi [A] time = 0.0087903, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2295, 2389} \[ -3 x+x \log (x)+(x+1) \log (x+1)+(x+2) \log (x+2) \]
Antiderivative was successfully verified.
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Rule 2295
Rule 2389
Rubi steps
\begin{align*} \int (\log (x)+\log (1+x)+\log (2+x)) \, dx &=\int \log (x) \, dx+\int \log (1+x) \, dx+\int \log (2+x) \, dx\\ &=-x+x \log (x)+\operatorname{Subst}(\int \log (x) \, dx,x,1+x)+\operatorname{Subst}(\int \log (x) \, dx,x,2+x)\\ &=-3 x+x \log (x)+(1+x) \log (1+x)+(2+x) \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0029841, size = 30, normalized size = 1.25 \[ -3 x+x \log (x)+x \log (x+1)+x \log (x+2)+\log (x+1)+2 \log (x+2) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 1.1 \begin{align*} x\ln \left ( x \right ) -3\,x+ \left ( 1+x \right ) \ln \left ( 1+x \right ) -3+ \left ( 2+x \right ) \ln \left ( 2+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946145, size = 34, normalized size = 1.42 \begin{align*}{\left (x + 2\right )} \log \left (x + 2\right ) +{\left (x + 1\right )} \log \left (x + 1\right ) + x \log \left (x\right ) - 3 \, x - 3 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57867, size = 78, normalized size = 3.25 \begin{align*}{\left (x + 2\right )} \log \left (x + 2\right ) +{\left (x + 1\right )} \log \left (x + 1\right ) + x \log \left (x\right ) - 3 \, x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.44325, size = 32, normalized size = 1.33 \begin{align*} x \log{\left (x \right )} + x \log{\left (x + 1 \right )} + x \log{\left (x + 2 \right )} - 3 x + \log{\left (x + 1 \right )} + 2 \log{\left (x + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08147, size = 34, normalized size = 1.42 \begin{align*}{\left (x + 2\right )} \log \left (x + 2\right ) +{\left (x + 1\right )} \log \left (x + 1\right ) + x \log \left (x\right ) - 3 \, x - 3 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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