\(\int x \log (x) \, dx\) [58]

Optimal result

Integrand size = 4, antiderivative size = 17 \[ \int x \log (x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2341} \[ \int x \log (x) \, dx=\frac {1}{2} x^2 \log (x)-\frac {x^2}{4} \]

[In]

Int[x*Log[x],x]

[Out]

-1/4*x^2 + (x^2*Log[x])/2

Rule 2341

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*} \text {integral}& = -\frac {x^2}{4}+\frac {1}{2} x^2 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int x \log (x) \, dx=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (x) \]

[In]

Integrate[x*Log[x],x]

[Out]

-1/4*x^2 + (x^2*Log[x])/2

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\)	\(14\)
norman	\(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\)	\(14\)
risch	\(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\)	\(14\)
parallelrisch	\(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\)	\(14\)
parts	\(-\frac {x^{2}}{4}+\frac {x^{2} \ln \left (x \right )}{2}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \log (x) \, dx=\frac {1}{2} \, x^{2} \log \left (x\right ) - \frac {1}{4} \, x^{2} \]

[In]

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

[Out]

1/2*x^2*log(x) - 1/4*x^2

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.71 \[ \int x \log (x) \, dx=\frac {x^{2} \log {\left (x \right )}}{2} - \frac {x^{2}}{4} \]

[In]

integrate(x*ln(x),x)

[Out]

x**2*log(x)/2 - x**2/4

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \log (x) \, dx=\frac {1}{2} \, x^{2} \log \left (x\right ) - \frac {1}{4} \, x^{2} \]

[In]

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

[Out]

1/2*x^2*log(x) - 1/4*x^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int x \log (x) \, dx=\frac {1}{2} \, x^{2} \log \left (x\right ) - \frac {1}{4} \, x^{2} \]

[In]

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

[Out]

1/2*x^2*log(x) - 1/4*x^2

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.53 \[ \int x \log (x) \, dx=\frac {x^2\,\left (\ln \left (x\right )-\frac {1}{2}\right )}{2} \]

[In]

int(x*log(x),x)

[Out]

(x^2*(log(x) - 1/2))/2