3.98.44 \(\int \frac {(6-30 x+5 x^2) \log (x)+(150 x^2-25 x^3+(30 x-5 x^2) \log (\frac {6-x}{x})) \log ^2(x)+(30 x-5 x^2+(6-x) \log (\frac {6-x}{x})) \log (5 x+\log (\frac {6-x}{x}))}{(150 x^3-25 x^4+(30 x^2-5 x^3) \log (\frac {6-x}{x})) \log ^2(x)+(-30 x^2+5 x^3+(-6 x+x^2) \log (\frac {6-x}{x})) \log (x) \log (5 x+\log (\frac {6-x}{x}))} \, dx\)

Optimal. Leaf size=25 \[ \log \left (-5 x+\frac {\log \left (5 x+\log \left (\frac {6-x}{x}\right )\right )}{\log (x)}\right ) \]

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Rubi [A]  time = 12.33, antiderivative size = 28, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, integrand size = 180, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6688, 6742, 2302, 29, 6684} \begin {gather*} \log \left (5 x \log (x)-\log \left (5 x+\log \left (\frac {6}{x}-1\right )\right )\right )-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((6 - 30*x + 5*x^2)*Log[x] + (150*x^2 - 25*x^3 + (30*x - 5*x^2)*Log[(6 - x)/x])*Log[x]^2 + (30*x - 5*x^2 +
(6 - x)*Log[(6 - x)/x])*Log[5*x + Log[(6 - x)/x]])/((150*x^3 - 25*x^4 + (30*x^2 - 5*x^3)*Log[(6 - x)/x])*Log[
x]^2 + (-30*x^2 + 5*x^3 + (-6*x + x^2)*Log[(6 - x)/x])*Log[x]*Log[5*x + Log[(6 - x)/x]]),x]

[Out]

-Log[Log[x]] + Log[5*x*Log[x] - Log[5*x + Log[-1 + 6/x]]]

Rule 29

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

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 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 {-\left (\left (-6+30 x-5 x^2\right ) \log (x)\right )-5 (-6+x) x \left (5 x+\log \left (-1+\frac {6}{x}\right )\right ) \log ^2(x)-(-6+x) \left (5 x+\log \left (-1+\frac {6}{x}\right )\right ) \log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )}{(6-x) x \left (5 x+\log \left (-1+\frac {6}{x}\right )\right ) \log (x) \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right )} \, dx\\ &=\int \left (-\frac {1}{x \log (x)}+\frac {-6+30 x-155 x^2+25 x^3-30 x \log \left (-1+\frac {6}{x}\right )+5 x^2 \log \left (-1+\frac {6}{x}\right )-150 x^2 \log (x)+25 x^3 \log (x)-30 x \log \left (-1+\frac {6}{x}\right ) \log (x)+5 x^2 \log \left (-1+\frac {6}{x}\right ) \log (x)}{(-6+x) x \left (5 x+\log \left (-1+\frac {6}{x}\right )\right ) \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right )}\right ) \, dx\\ &=-\int \frac {1}{x \log (x)} \, dx+\int \frac {-6+30 x-155 x^2+25 x^3-30 x \log \left (-1+\frac {6}{x}\right )+5 x^2 \log \left (-1+\frac {6}{x}\right )-150 x^2 \log (x)+25 x^3 \log (x)-30 x \log \left (-1+\frac {6}{x}\right ) \log (x)+5 x^2 \log \left (-1+\frac {6}{x}\right ) \log (x)}{(-6+x) x \left (5 x+\log \left (-1+\frac {6}{x}\right )\right ) \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right )} \, dx\\ &=\log \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-\log (\log (x))+\log \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 28, normalized size = 1.12 \begin {gather*} -\log (\log (x))+\log \left (5 x \log (x)-\log \left (5 x+\log \left (-1+\frac {6}{x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((6 - 30*x + 5*x^2)*Log[x] + (150*x^2 - 25*x^3 + (30*x - 5*x^2)*Log[(6 - x)/x])*Log[x]^2 + (30*x - 5
*x^2 + (6 - x)*Log[(6 - x)/x])*Log[5*x + Log[(6 - x)/x]])/((150*x^3 - 25*x^4 + (30*x^2 - 5*x^3)*Log[(6 - x)/x]
)*Log[x]^2 + (-30*x^2 + 5*x^3 + (-6*x + x^2)*Log[(6 - x)/x])*Log[x]*Log[5*x + Log[(6 - x)/x]]),x]

[Out]

-Log[Log[x]] + Log[5*x*Log[x] - Log[5*x + Log[-1 + 6/x]]]

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fricas [A]  time = 0.94, size = 27, normalized size = 1.08 \begin {gather*} \log \left (-5 \, x \log \relax (x) + \log \left (5 \, x + \log \left (-\frac {x - 6}{x}\right )\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+6)*log((-x+6)/x)-5*x^2+30*x)*log(log((-x+6)/x)+5*x)+((-5*x^2+30*x)*log((-x+6)/x)-25*x^3+150*x^
2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2-6*x)*log((-x+6)/x)+5*x^3-30*x^2)*log(x)*log(log((-x+6)/x)+5*x)+((-5*
x^3+30*x^2)*log((-x+6)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="fricas")

[Out]

log(-5*x*log(x) + log(5*x + log(-(x - 6)/x))) - log(log(x))

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giac [A]  time = 0.75, size = 28, normalized size = 1.12 \begin {gather*} \log \left (-5 \, x \log \relax (x) + \log \left (5 \, x - \log \relax (x) + \log \left (-x + 6\right )\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+6)*log((-x+6)/x)-5*x^2+30*x)*log(log((-x+6)/x)+5*x)+((-5*x^2+30*x)*log((-x+6)/x)-25*x^3+150*x^
2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2-6*x)*log((-x+6)/x)+5*x^3-30*x^2)*log(x)*log(log((-x+6)/x)+5*x)+((-5*
x^3+30*x^2)*log((-x+6)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="giac")

[Out]

log(-5*x*log(x) + log(5*x - log(x) + log(-x + 6))) - log(log(x))

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maple [C]  time = 0.31, size = 113, normalized size = 4.52

method	result	size
risch	\(-\ln \left (\ln \relax (x )\right )+\ln \left (-5 x \ln \relax (x )+\ln \left (i \pi -\ln \relax (x )+\ln \left (x -6\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \left (x -6\right )}{x}\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -6\right )}{x}\right )+\mathrm {csgn}\left (\frac {i}{x}\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \left (x -6\right )}{x}\right )+\mathrm {csgn}\left (i \left (x -6\right )\right )\right )}{2}+i \pi \mathrm {csgn}\left (\frac {i \left (x -6\right )}{x}\right )^{2} \left (\mathrm {csgn}\left (\frac {i \left (x -6\right )}{x}\right )-1\right )+5 x \right )\right )\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x+6)*ln((-x+6)/x)-5*x^2+30*x)*ln(ln((-x+6)/x)+5*x)+((-5*x^2+30*x)*ln((-x+6)/x)-25*x^3+150*x^2)*ln(x)^2
+(5*x^2-30*x+6)*ln(x))/(((x^2-6*x)*ln((-x+6)/x)+5*x^3-30*x^2)*ln(x)*ln(ln((-x+6)/x)+5*x)+((-5*x^3+30*x^2)*ln((
-x+6)/x)-25*x^4+150*x^3)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x))+ln(-5*x*ln(x)+ln(I*Pi-ln(x)+ln(x-6)-1/2*I*Pi*csgn(I/x*(x-6))*(-csgn(I/x*(x-6))+csgn(I/x))*(-csgn(I/
x*(x-6))+csgn(I*(x-6)))+I*Pi*csgn(I/x*(x-6))^2*(csgn(I/x*(x-6))-1)+5*x))

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maxima [A]  time = 0.49, size = 28, normalized size = 1.12 \begin {gather*} \log \left (-5 \, x \log \relax (x) + \log \left (5 \, x - \log \relax (x) + \log \left (-x + 6\right )\right )\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+6)*log((-x+6)/x)-5*x^2+30*x)*log(log((-x+6)/x)+5*x)+((-5*x^2+30*x)*log((-x+6)/x)-25*x^3+150*x^
2)*log(x)^2+(5*x^2-30*x+6)*log(x))/(((x^2-6*x)*log((-x+6)/x)+5*x^3-30*x^2)*log(x)*log(log((-x+6)/x)+5*x)+((-5*
x^3+30*x^2)*log((-x+6)/x)-25*x^4+150*x^3)*log(x)^2),x, algorithm="maxima")

[Out]

log(-5*x*log(x) + log(5*x - log(x) + log(-x + 6))) - log(log(x))

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mupad [B]  time = 6.64, size = 27, normalized size = 1.08 \begin {gather*} \ln \left (\ln \left (5\,x+\ln \left (-\frac {x-6}{x}\right )\right )-5\,x\,\ln \relax (x)\right )-\ln \left (\ln \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(5*x^2 - 30*x + 6) - log(5*x + log(-(x - 6)/x))*(log(-(x - 6)/x)*(x - 6) - 30*x + 5*x^2) + log(x)^
2*(log(-(x - 6)/x)*(30*x - 5*x^2) + 150*x^2 - 25*x^3))/(log(x)^2*(log(-(x - 6)/x)*(30*x^2 - 5*x^3) + 150*x^3 -
25*x^4) - log(5*x + log(-(x - 6)/x))*log(x)*(log(-(x - 6)/x)*(6*x - x^2) + 30*x^2 - 5*x^3)),x)

[Out]

log(log(5*x + log(-(x - 6)/x)) - 5*x*log(x)) - log(log(x))

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sympy [A]  time = 1.53, size = 24, normalized size = 0.96 \begin {gather*} \log {\left (- 5 x \log {\relax (x )} + \log {\left (5 x + \log {\left (\frac {6 - x}{x} \right )} \right )} \right )} - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x+6)*ln((-x+6)/x)-5*x**2+30*x)*ln(ln((-x+6)/x)+5*x)+((-5*x**2+30*x)*ln((-x+6)/x)-25*x**3+150*x**
2)*ln(x)**2+(5*x**2-30*x+6)*ln(x))/(((x**2-6*x)*ln((-x+6)/x)+5*x**3-30*x**2)*ln(x)*ln(ln((-x+6)/x)+5*x)+((-5*x
**3+30*x**2)*ln((-x+6)/x)-25*x**4+150*x**3)*ln(x)**2),x)

[Out]

log(-5*x*log(x) + log(5*x + log((6 - x)/x))) - log(log(x))

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