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

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

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Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2305, 2304} \[ \frac {x^2}{4}+\frac {1}{2} x^2 \log ^2(x)-\frac {1}{2} x^2 \log (x) \]

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

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \[ \frac {x^2}{4}+\frac {1}{2} x^2 \log ^2(x)-\frac {1}{2} x^2 \log (x) \]

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log ^2(x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[x*Log[x]^2,x]

[Out]

Could not integrate

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fricas [A]  time = 0.92, size = 22, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} \log \relax (x)^{2} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(x)^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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giac [A]  time = 0.88, size = 22, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} \log \relax (x)^{2} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 23, normalized size = 0.82

method	result	size
default	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \relax (x )}{2}+\frac {x^{2} \ln \relax (x )^{2}}{2}\)	\(23\)
norman	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \relax (x )}{2}+\frac {x^{2} \ln \relax (x )^{2}}{2}\)	\(23\)
risch	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \relax (x )}{2}+\frac {x^{2} \ln \relax (x )^{2}}{2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(x)^2,x,method=_RETURNVERBOSE)

[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 = 0.44, size = 17, normalized size = 0.61 \[ \frac {1}{4} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} \]

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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mupad [B]  time = 0.03, size = 17, normalized size = 0.61 \[ \frac {x^2\,\left (2\,{\ln \relax (x)}^2-2\,\ln \relax (x)+1\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(x)^2,x)

[Out]

(x^2*(2*log(x)^2 - 2*log(x) + 1))/4

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sympy [A]  time = 0.11, size = 22, normalized size = 0.79 \[ \frac {x^{2} \log {\relax (x )}^{2}}{2} - \frac {x^{2} \log {\relax (x )}}{2} + \frac {x^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(x)**2,x)

[Out]

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

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