3.27.21 \(\int \frac {-3+x \log (x)}{x} \, dx\)

Optimal. Leaf size=11 \[ -3-x+(-3+x) \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 2295} \begin {gather*} -x+x \log (x)-3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + x*Log[x])/x,x]

[Out]

-x - 3*Log[x] + x*Log[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} &=\int \left (-\frac {3}{x}+\log (x)\right ) \, dx\\ &=-3 \log (x)+\int \log (x) \, dx\\ &=-x-3 \log (x)+x \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} -x-3 \log (x)+x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + x*Log[x])/x,x]

[Out]

-x - 3*Log[x] + x*Log[x]

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fricas [A]  time = 0.74, size = 10, normalized size = 0.91 \begin {gather*} {\left (x - 3\right )} \log \relax (x) - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x - 3)*log(x) - x

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giac [A]  time = 0.19, size = 12, normalized size = 1.09 \begin {gather*} x \log \relax (x) - x - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x) - x - 3*log(x)

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maple [A]  time = 0.01, size = 13, normalized size = 1.18

method	result	size
default	\(x \ln \relax (x )-x -3 \ln \relax (x )\)	\(13\)
norman	\(x \ln \relax (x )-x -3 \ln \relax (x )\)	\(13\)
risch	\(x \ln \relax (x )-x -3 \ln \relax (x )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*ln(x)-x-3*ln(x)

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maxima [A]  time = 0.63, size = 12, normalized size = 1.09 \begin {gather*} x \log \relax (x) - x - 3 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log(x) - x - 3*log(x)

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mupad [B]  time = 1.41, size = 12, normalized size = 1.09 \begin {gather*} x\,\ln \relax (x)-3\,\ln \relax (x)-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(x) - 3)/x,x)

[Out]

x*log(x) - 3*log(x) - x

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sympy [A]  time = 0.09, size = 10, normalized size = 0.91 \begin {gather*} x \log {\relax (x )} - x - 3 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x*ln(x)-3)/x,x)

[Out]

x*log(x) - x - 3*log(x)

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