3.34.53 \(\int \frac {-5+x-x \log (x)}{(-5 x+x^2) \log (x)} \, dx\)

Optimal. Leaf size=17 \[ \log \left (\frac {5 \log (2) \log (x)}{6 e (-5+x)}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 12, normalized size of antiderivative = 0.71, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 6742, 2302, 29} \begin {gather*} \log (\log (x))-\log (5-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + x - x*Log[x])/((-5*x + x^2)*Log[x]),x]

[Out]

-Log[5 - x] + Log[Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5+x-x \log (x)}{(-5+x) x \log (x)} \, dx\\ &=\int \left (\frac {1}{5-x}+\frac {1}{x \log (x)}\right ) \, dx\\ &=-\log (5-x)+\int \frac {1}{x \log (x)} \, dx\\ &=-\log (5-x)+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (5-x)+\log (\log (x))\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 12, normalized size = 0.71 \begin {gather*} -\log (5-x)+\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + x - x*Log[x])/((-5*x + x^2)*Log[x]),x]

[Out]

-Log[5 - x] + Log[Log[x]]

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 10, normalized size = 0.59 \begin {gather*} -\log \left (x - 5\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+x-5)/(x^2-5*x)/log(x),x, algorithm="fricas")

[Out]

-log(x - 5) + log(log(x))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 10, normalized size = 0.59 \begin {gather*} -\log \left (x - 5\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+x-5)/(x^2-5*x)/log(x),x, algorithm="giac")

[Out]

-log(x - 5) + log(log(x))

________________________________________________________________________________________

maple [A]  time = 0.46, size = 11, normalized size = 0.65

method	result	size
default	\(\ln \left (\ln \relax (x )\right )-\ln \left (x -5\right )\)	\(11\)
norman	\(\ln \left (\ln \relax (x )\right )-\ln \left (x -5\right )\)	\(11\)
risch	\(\ln \left (\ln \relax (x )\right )-\ln \left (x -5\right )\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x)+x-5)/(x^2-5*x)/ln(x),x,method=_RETURNVERBOSE)

[Out]

ln(ln(x))-ln(x-5)

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 10, normalized size = 0.59 \begin {gather*} -\log \left (x - 5\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)+x-5)/(x^2-5*x)/log(x),x, algorithm="maxima")

[Out]

-log(x - 5) + log(log(x))

________________________________________________________________________________________

mupad [B]  time = 2.04, size = 10, normalized size = 0.59 \begin {gather*} \ln \left (\ln \relax (x)\right )-\ln \left (x-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*log(x) - x + 5)/(log(x)*(5*x - x^2)),x)

[Out]

log(log(x)) - log(x - 5)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 8, normalized size = 0.47 \begin {gather*} - \log {\left (x - 5 \right )} + \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x)+x-5)/(x**2-5*x)/ln(x),x)

[Out]

-log(x - 5) + log(log(x))

________________________________________________________________________________________