3.30.59 \(\int \frac {3-4 x \log (x)}{4 x} \, dx\)

Optimal. Leaf size=11 \[ x-\left (-\frac {3}{4}+x\right ) \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 14, 2295} \begin {gather*} x+x (-\log (x))+\frac {3 \log (x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x*Log[x])/(4*x),x]

[Out]

x + (3*Log[x])/4 - x*Log[x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {3-4 x \log (x)}{x} \, dx\\ &=\frac {1}{4} \int \left (\frac {3}{x}-4 \log (x)\right ) \, dx\\ &=\frac {3 \log (x)}{4}-\int \log (x) \, dx\\ &=x+\frac {3 \log (x)}{4}-x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.18 \begin {gather*} x+\frac {3 \log (x)}{4}-x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x*Log[x])/(4*x),x]

[Out]

x + (3*Log[x])/4 - x*Log[x]

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fricas [A]  time = 0.56, size = 11, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, {\left (4 \, x - 3\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)+3)/x,x, algorithm="fricas")

[Out]

-1/4*(4*x - 3)*log(x) + x

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giac [A]  time = 0.21, size = 11, normalized size = 1.00 \begin {gather*} -x \log \relax (x) + x + \frac {3}{4} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)+3)/x,x, algorithm="giac")

[Out]

-x*log(x) + x + 3/4*log(x)

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maple [A]  time = 0.02, size = 12, normalized size = 1.09

method	result	size
default	\(-x \ln \relax (x )+x +\frac {3 \ln \relax (x )}{4}\)	\(12\)
norman	\(-x \ln \relax (x )+x +\frac {3 \ln \relax (x )}{4}\)	\(12\)
risch	\(-x \ln \relax (x )+x +\frac {3 \ln \relax (x )}{4}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*x*ln(x)+3)/x,x,method=_RETURNVERBOSE)

[Out]

-x*ln(x)+x+3/4*ln(x)

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maxima [A]  time = 0.36, size = 11, normalized size = 1.00 \begin {gather*} -x \log \relax (x) + x + \frac {3}{4} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*log(x)+3)/x,x, algorithm="maxima")

[Out]

-x*log(x) + x + 3/4*log(x)

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mupad [B]  time = 1.70, size = 11, normalized size = 1.00 \begin {gather*} x+\frac {3\,\ln \relax (x)}{4}-x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*log(x) - 3/4)/x,x)

[Out]

x + (3*log(x))/4 - x*log(x)

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sympy [A]  time = 0.09, size = 12, normalized size = 1.09 \begin {gather*} - x \log {\relax (x )} + x + \frac {3 \log {\relax (x )}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*x*ln(x)+3)/x,x)

[Out]

-x*log(x) + x + 3*log(x)/4

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