3.58 \(\int x \log (x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {1}{2} x^2 \log (x)-\frac {x^2}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[x],x]

[Out]

-1/4*x^2 + (x^2*Log[x])/2

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fricas [A]  time = 0.39, size = 13, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} \log \relax (x) - \frac {1}{4} \, x^{2} \]

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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giac [A]  time = 0.01, size = 13, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} \log \relax (x) - \frac {1}{4} \, x^{2} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {x^{2} \ln \relax (x )}{2}-\frac {x^{2}}{4} \]

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 = 0.50, size = 13, normalized size = 0.76 \[ \frac {1}{2} \, x^{2} \log \relax (x) - \frac {1}{4} \, x^{2} \]

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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mupad [B]  time = 0.03, size = 9, normalized size = 0.53 \[ \frac {x^2\,\left (\ln \relax (x)-\frac {1}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(x),x)

[Out]

(x^2*(log(x) - 1/2))/2

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sympy [A]  time = 0.09, size = 12, normalized size = 0.71 \[ \frac {x^{2} \log {\relax (x )}}{2} - \frac {x^{2}}{4} \]

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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