### 3.23 $$\int x \log (x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{2} x^2 \log (x)-\frac{x^2}{4}$

[Out]

-x^2/4 + (x^2*Log[x])/2

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Rubi [A]  time = 0.0034969, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 4, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {2304} $\frac{1}{2} x^2 \log (x)-\frac{x^2}{4}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Log[x],x]

[Out]

-x^2/4 + (x^2*Log[x])/2

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int x \log (x) \, dx &=-\frac{x^2}{4}+\frac{1}{2} x^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.000605, size = 17, normalized size = 1. $\frac{1}{2} x^2 \log (x)-\frac{x^2}{4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x*Log[x],x]

[Out]

-x^2/4 + (x^2*Log[x])/2

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Maple [A]  time = 0., size = 14, normalized size = 0.8 \begin{align*} -{\frac{{x}^{2}}{4}}+{\frac{{x}^{2}\ln \left ( x \right ) }{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(x),x)

[Out]

-1/4*x^2+1/2*x^2*ln(x)

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Maxima [A]  time = 0.935179, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x\right ) - \frac{1}{4} \, x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x),x, algorithm="maxima")

[Out]

1/2*x^2*log(x) - 1/4*x^2

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Fricas [A]  time = 1.87314, size = 35, normalized size = 2.06 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x\right ) - \frac{1}{4} \, x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x),x, algorithm="fricas")

[Out]

1/2*x^2*log(x) - 1/4*x^2

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Sympy [A]  time = 0.080408, size = 12, normalized size = 0.71 \begin{align*} \frac{x^{2} \log{\left (x \right )}}{2} - \frac{x^{2}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(x),x)

[Out]

x**2*log(x)/2 - x**2/4

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Giac [A]  time = 1.05056, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{2} \, x^{2} \log \left (x\right ) - \frac{1}{4} \, x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x),x, algorithm="giac")

[Out]

1/2*x^2*log(x) - 1/4*x^2