\(\int \frac {-3+x \log (x)}{x} \, dx\) [2621]

Optimal result

Integrand size = 10, antiderivative size = 11 \[ \int \frac {-3+x \log (x)}{x} \, dx=-3-x+(-3+x) \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {14, 2332} \[ \int \frac {-3+x \log (x)}{x} \, dx=-x+x \log (x)-3 \log (x) \]

[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 2332

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {-3+x \log (x)}{x} \, dx=-x-3 \log (x)+x \log (x) \]

[In]

Integrate[(-3 + x*Log[x])/x,x]

[Out]

-x - 3*Log[x] + x*Log[x]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18

method	result	size
default	\(x \ln \left (x \right )-x -3 \ln \left (x \right )\)	\(13\)
norman	\(x \ln \left (x \right )-x -3 \ln \left (x \right )\)	\(13\)
risch	\(x \ln \left (x \right )-x -3 \ln \left (x \right )\)	\(13\)
parallelrisch	\(x \ln \left (x \right )-x -3 \ln \left (x \right )\)	\(13\)
parts	\(x \ln \left (x \right )-x -3 \ln \left (x \right )\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {-3+x \log (x)}{x} \, dx={\left (x - 3\right )} \log \left (x\right ) - x \]

[In]

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

[Out]

(x - 3)*log(x) - x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91 \[ \int \frac {-3+x \log (x)}{x} \, dx=x \log {\left (x \right )} - x - 3 \log {\left (x \right )} \]

[In]

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

[Out]

x*log(x) - x - 3*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {-3+x \log (x)}{x} \, dx=x \log \left (x\right ) - x - 3 \, \log \left (x\right ) \]

[In]

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

[Out]

x*log(x) - x - 3*log(x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {-3+x \log (x)}{x} \, dx=x \log \left (x\right ) - x - 3 \, \log \left (x\right ) \]

[In]

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

[Out]

x*log(x) - x - 3*log(x)

Mupad [B] (verification not implemented)

Time = 8.46 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {-3+x \log (x)}{x} \, dx=x\,\ln \left (x\right )-3\,\ln \left (x\right )-x \]

[In]

int((x*log(x) - 3)/x,x)

[Out]

x*log(x) - 3*log(x) - x