3.64 \(\int x \log ^2(x) \, dx\)

Optimal. Leaf size=28 \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]

[Out]

x^2/4 - (x^2*Log[x])/2 + (x^2*Log[x]^2)/2

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Rubi [A]  time = 0.016315, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[x*Log[x]^2,x]

[Out]

x^2/4 - (x^2*Log[x])/2 + (x^2*Log[x]^2)/2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*ln(x)**2,x)

[Out]

x**2*log(x)**2/2 - x**2*log(x)/2 + Integral(x, x)/2

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Mathematica [A]  time = 0.00186486, size = 28, normalized size = 1. \[ \frac{x^2}{4}+\frac{1}{2} x^2 \log ^2(x)-\frac{1}{2} x^2 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x*Log[x]^2,x]

[Out]

x^2/4 - (x^2*Log[x])/2 + (x^2*Log[x]^2)/2

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Maple [A]  time = 0., size = 23, normalized size = 0.8 \[{\frac{{x}^{2}}{4}}-{\frac{{x}^{2}\ln \left ( x \right ) }{2}}+{\frac{{x}^{2} \left ( \ln \left ( x \right ) \right ) ^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*ln(x)^2,x)

[Out]

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

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Maxima [A]  time = 1.3461, size = 23, normalized size = 0.82 \[ \frac{1}{4} \,{\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x)^2,x, algorithm="maxima")

[Out]

1/4*(2*log(x)^2 - 2*log(x) + 1)*x^2

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Fricas [A]  time = 0.225426, size = 30, normalized size = 1.07 \[ \frac{1}{2} \, x^{2} \log \left (x\right )^{2} - \frac{1}{2} \, x^{2} \log \left (x\right ) + \frac{1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.092948, size = 22, normalized size = 0.79 \[ \frac{x^{2} \log{\left (x \right )}^{2}}{2} - \frac{x^{2} \log{\left (x \right )}}{2} + \frac{x^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*ln(x)**2,x)

[Out]

x**2*log(x)**2/2 - x**2*log(x)/2 + x**2/4

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GIAC/XCAS [A]  time = 0.210396, size = 30, normalized size = 1.07 \[ \frac{1}{2} \, x^{2}{\rm ln}\left (x\right )^{2} - \frac{1}{2} \, x^{2}{\rm ln}\left (x\right ) + \frac{1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*log(x)^2,x, algorithm="giac")

[Out]

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