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 = 0.00772183, 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 \[ \frac{1}{2} x^2 \log (x)-\frac{x^2}{4} \]

Antiderivative was successfully verified.

[In]  Int[x*Log[x],x]

[Out]

-x^2/4 + (x^2*Log[x])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{2} \log{\left (x \right )}}{2} - \frac{\int x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*ln(x),x)

[Out]

x**2*log(x)/2 - Integral(x, x)/2

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Mathematica [A]  time = 0.00118458, 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 \[ -{\frac{{x}^{2}}{4}}+{\frac{{x}^{2}\ln \left ( x \right ) }{2}} \]

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 = 1.33885, size = 18, normalized size = 1.06 \[ \frac{1}{2} \, x^{2} \log \left (x\right ) - \frac{1}{4} \, x^{2} \]

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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Fricas [A]  time = 0.216551, size = 18, normalized size = 1.06 \[ \frac{1}{2} \, x^{2} \log \left (x\right ) - \frac{1}{4} \, x^{2} \]

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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Sympy [A]  time = 0.069775, size = 12, normalized size = 0.71 \[ \frac{x^{2} \log{\left (x \right )}}{2} - \frac{x^{2}}{4} \]

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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GIAC/XCAS [A]  time = 0.201276, size = 18, normalized size = 1.06 \[ \frac{1}{2} \, x^{2}{\rm ln}\left (x\right ) - \frac{1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x),x, algorithm="giac")

[Out]

1/2*x^2*ln(x) - 1/4*x^2