3.1.78 \(\int \frac {-4 x-\log (x)}{6 x} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{3} \left (5-2 x-\frac {\log ^2(x)}{4}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 14, 2301} \begin {gather*} -\frac {2 x}{3}-\frac {1}{12} \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x - Log[x])/(6*x),x]

[Out]

(-2*x)/3 - Log[x]^2/12

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{6} \int \frac {-4 x-\log (x)}{x} \, dx\\ &=\frac {1}{6} \int \left (-4-\frac {\log (x)}{x}\right ) \, dx\\ &=-\frac {2 x}{3}-\frac {1}{6} \int \frac {\log (x)}{x} \, dx\\ &=-\frac {2 x}{3}-\frac {\log ^2(x)}{12}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} -\frac {2 x}{3}-\frac {\log ^2(x)}{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x - Log[x])/(6*x),x]

[Out]

(-2*x)/3 - Log[x]^2/12

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 10, normalized size = 0.59 \begin {gather*} -\frac {1}{12} \, \log \relax (x)^{2} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-log(x)-4*x)/x,x, algorithm="fricas")

[Out]

-1/12*log(x)^2 - 2/3*x

________________________________________________________________________________________

giac [A]  time = 0.25, size = 10, normalized size = 0.59 \begin {gather*} -\frac {1}{12} \, \log \relax (x)^{2} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-log(x)-4*x)/x,x, algorithm="giac")

[Out]

-1/12*log(x)^2 - 2/3*x

________________________________________________________________________________________

maple [A]  time = 0.01, size = 11, normalized size = 0.65

method	result	size
default	\(-\frac {2 x}{3}-\frac {\ln \relax (x )^{2}}{12}\)	\(11\)
norman	\(-\frac {2 x}{3}-\frac {\ln \relax (x )^{2}}{12}\)	\(11\)
risch	\(-\frac {2 x}{3}-\frac {\ln \relax (x )^{2}}{12}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6*(-ln(x)-4*x)/x,x,method=_RETURNVERBOSE)

[Out]

-2/3*x-1/12*ln(x)^2

________________________________________________________________________________________

maxima [A]  time = 0.80, size = 10, normalized size = 0.59 \begin {gather*} -\frac {1}{12} \, \log \relax (x)^{2} - \frac {2}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-log(x)-4*x)/x,x, algorithm="maxima")

[Out]

-1/12*log(x)^2 - 2/3*x

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 10, normalized size = 0.59 \begin {gather*} -\frac {{\ln \relax (x)}^2}{12}-\frac {2\,x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x)/3 + log(x)/6)/x,x)

[Out]

- (2*x)/3 - log(x)^2/12

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 12, normalized size = 0.71 \begin {gather*} - \frac {2 x}{3} - \frac {\log {\relax (x )}^{2}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(-ln(x)-4*x)/x,x)

[Out]

-2*x/3 - log(x)**2/12

________________________________________________________________________________________