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

Optimal. Leaf size=12 \[ \log \left (\frac {16 (3-\log (x))}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {43} \begin {gather*} \log (3-\log (x))-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - Log[x])/(-3*x + x*Log[x]),x]

[Out]

-Log[x] + Log[3 - Log[x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 10, normalized size = 0.83 \begin {gather*} -\log (x)+\log (-3+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 - Log[x])/(-3*x + x*Log[x]),x]

[Out]

-Log[x] + Log[-3 + Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.58, size = 10, normalized size = 0.83 \begin {gather*} -\log \relax (x) + \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(log(x) - 3)

________________________________________________________________________________________

giac [B]  time = 0.33, size = 27, normalized size = 2.25 \begin {gather*} \frac {1}{2} \, \log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\relax (x) - 1\right )}^{2} + {\left (\log \left ({\left | x \right |}\right ) - 3\right )}^{2}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(1/4*pi^2*(sgn(x) - 1)^2 + (log(abs(x)) - 3)^2) - log(x)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 11, normalized size = 0.92

method	result	size
default	\(-\ln \relax (x )+\ln \left (\ln \relax (x )-3\right )\)	\(11\)
norman	\(-\ln \relax (x )+\ln \left (\ln \relax (x )-3\right )\)	\(11\)
risch	\(-\ln \relax (x )+\ln \left (\ln \relax (x )-3\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.38, size = 32, normalized size = 2.67 \begin {gather*} {\left (\log \relax (x) - 3\right )} \log \left (\log \relax (x) - 3\right ) - \log \relax (x) \log \left (\log \relax (x) - 3\right ) - \log \relax (x) + 4 \, \log \left (\log \relax (x) - 3\right ) + 3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(x) - 3)*log(log(x) - 3) - log(x)*log(log(x) - 3) - log(x) + 4*log(log(x) - 3) + 3

________________________________________________________________________________________

mupad [B]  time = 0.46, size = 10, normalized size = 0.83 \begin {gather*} \ln \left (\ln \relax (x)-3\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) - 4)/(3*x - x*log(x)),x)

[Out]

log(log(x) - 3) - log(x)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 8, normalized size = 0.67 \begin {gather*} - \log {\relax (x )} + \log {\left (\log {\relax (x )} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x) + log(log(x) - 3)

________________________________________________________________________________________