3.3 \(\int x \log (c x) \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{2} x^2 \log (c x)-\frac {x^2}{4} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2304} \[ \frac {1}{2} x^2 \log (c x)-\frac {x^2}{4} \]

Antiderivative was successfully verified.

[In]

Int[x*Log[c*x],x]

[Out]

-x^2/4 + (x^2*Log[c*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 (c x) \, dx &=-\frac {x^2}{4}+\frac {1}{2} x^2 \log (c x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \[ \frac {1}{2} x^2 \log (c x)-\frac {x^2}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Log[c*x],x]

[Out]

-1/4*x^2 + (x^2*Log[c*x])/2

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fricas [A]  time = 0.42, size = 15, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(c*x) - 1/4*x^2

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giac [A]  time = 0.22, size = 15, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(c*x) - 1/4*x^2

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maple [A]  time = 0.03, size = 16, normalized size = 0.84 \[ \frac {x^{2} \ln \left (c x \right )}{2}-\frac {x^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*ln(c*x),x)

[Out]

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

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maxima [A]  time = 0.56, size = 15, normalized size = 0.79 \[ \frac {1}{2} \, x^{2} \log \left (c x\right ) - \frac {1}{4} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*log(c*x) - 1/4*x^2

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mupad [B]  time = 0.03, size = 11, normalized size = 0.58 \[ \frac {x^2\,\left (\ln \left (c\,x\right )-\frac {1}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(c*x),x)

[Out]

(x^2*(log(c*x) - 1/2))/2

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sympy [A]  time = 0.10, size = 14, normalized size = 0.74 \[ \frac {x^{2} \log {\left (c x \right )}}{2} - \frac {x^{2}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(c*x),x)

[Out]

x**2*log(c*x)/2 - x**2/4

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