\(\int \frac {54-6 x^2+(72 x+8 x^2+8 x^3) \log (\frac {9+x+x^2}{x}) \log ^2(\log (\frac {9+x+x^2}{x}))}{((27 x+3 x^2+3 x^3) \log (\frac {9+x+x^2}{x}) \log (\log (\frac {9+x+x^2}{x}))+(-45 x+31 x^2-x^3+4 x^4) \log (\frac {9+x+x^2}{x}) \log ^2(\log (\frac {9+x+x^2}{x}))) \log (\frac {3+(-5+4 x) \log (\log (\frac {9+x+x^2}{x}))}{\log (\log (\frac {9+x+x^2}{x}))})} \, dx\) [4330]

Optimal result

Integrand size = 172, antiderivative size = 23 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=\log \left (\log ^2\left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )\right ) \]

[Out]

ln(ln(4*x+3/ln(ln(x+9/x+1))-5)^2)

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6820, 6816} \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \log \left (\log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )\right ) \]

[In]

Int[(54 - 6*x^2 + (72*x + 8*x^2 + 8*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]]^2)/(((27*x + 3*x^2 + 3
*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]] + (-45*x + 31*x^2 - x^3 + 4*x^4)*Log[(9 + x + x^2)/x]*Log
[Log[(9 + x + x^2)/x]]^2)*Log[(3 + (-5 + 4*x)*Log[Log[(9 + x + x^2)/x]])/Log[Log[(9 + x + x^2)/x]]]),x]

[Out]

2*Log[Log[-5 + 4*x + 3/Log[Log[1 + 9/x + x]]]]

Rule 6816

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

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6 \left (-9+x^2\right )+8 x \left (9+x+x^2\right ) \log \left (1+\frac {9}{x}+x\right ) \log ^2\left (\log \left (1+\frac {9}{x}+x\right )\right )}{x \left (9+x+x^2\right ) \log \left (1+\frac {9}{x}+x\right ) \log \left (\log \left (1+\frac {9}{x}+x\right )\right ) \left (3+(-5+4 x) \log \left (\log \left (1+\frac {9}{x}+x\right )\right )\right ) \log \left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )} \, dx \\ & = 2 \log \left (\log \left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \log \left (\log \left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )\right ) \]

[In]

Integrate[(54 - 6*x^2 + (72*x + 8*x^2 + 8*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]]^2)/(((27*x + 3*x
^2 + 3*x^3)*Log[(9 + x + x^2)/x]*Log[Log[(9 + x + x^2)/x]] + (-45*x + 31*x^2 - x^3 + 4*x^4)*Log[(9 + x + x^2)/
x]*Log[Log[(9 + x + x^2)/x]]^2)*Log[(3 + (-5 + 4*x)*Log[Log[(9 + x + x^2)/x]])/Log[Log[(9 + x + x^2)/x]]]),x]

[Out]

2*Log[Log[-5 + 4*x + 3/Log[Log[1 + 9/x + x]]]]

Maple [F]

\[\int \frac {\left (8 x^{3}+8 x^{2}+72 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) {\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}^{2}-6 x^{2}+54}{\left (\left (4 x^{4}-x^{3}+31 x^{2}-45 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) {\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}^{2}+\left (3 x^{3}+3 x^{2}+27 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) \ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )\right ) \ln \left (\frac {\left (-5+4 x \right ) \ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )+3}{\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}\right )}d x\]

[In]

int(((8*x^3+8*x^2+72*x)*ln((x^2+x+9)/x)*ln(ln((x^2+x+9)/x))^2-6*x^2+54)/((4*x^4-x^3+31*x^2-45*x)*ln((x^2+x+9)/
x)*ln(ln((x^2+x+9)/x))^2+(3*x^3+3*x^2+27*x)*ln((x^2+x+9)/x)*ln(ln((x^2+x+9)/x)))/ln(((-5+4*x)*ln(ln((x^2+x+9)/
x))+3)/ln(ln((x^2+x+9)/x))),x)

[Out]

int(((8*x^3+8*x^2+72*x)*ln((x^2+x+9)/x)*ln(ln((x^2+x+9)/x))^2-6*x^2+54)/((4*x^4-x^3+31*x^2-45*x)*ln((x^2+x+9)/
x)*ln(ln((x^2+x+9)/x))^2+(3*x^3+3*x^2+27*x)*ln((x^2+x+9)/x)*ln(ln((x^2+x+9)/x)))/ln(((-5+4*x)*ln(ln((x^2+x+9)/
x))+3)/ln(ln((x^2+x+9)/x))),x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \, \log \left (\log \left (\frac {{\left (4 \, x - 5\right )} \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right ) + 3}{\log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )}\right )\right ) \]

[In]

integrate(((8*x^3+8*x^2+72*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x))^2-6*x^2+54)/((4*x^4-x^3+31*x^2-45*x)*log(
(x^2+x+9)/x)*log(log((x^2+x+9)/x))^2+(3*x^3+3*x^2+27*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x)))/log(((-5+4*x)*
log(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm="fricas")

[Out]

2*log(log(((4*x - 5)*log(log((x^2 + x + 9)/x)) + 3)/log(log((x^2 + x + 9)/x))))

Sympy [A] (verification not implemented)

Time = 3.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \log {\left (\log {\left (\frac {\left (4 x - 5\right ) \log {\left (\log {\left (\frac {x^{2} + x + 9}{x} \right )} \right )} + 3}{\log {\left (\log {\left (\frac {x^{2} + x + 9}{x} \right )} \right )}} \right )} \right )} \]

[In]

integrate(((8*x**3+8*x**2+72*x)*ln((x**2+x+9)/x)*ln(ln((x**2+x+9)/x))**2-6*x**2+54)/((4*x**4-x**3+31*x**2-45*x
)*ln((x**2+x+9)/x)*ln(ln((x**2+x+9)/x))**2+(3*x**3+3*x**2+27*x)*ln((x**2+x+9)/x)*ln(ln((x**2+x+9)/x)))/ln(((-5
+4*x)*ln(ln((x**2+x+9)/x))+3)/ln(ln((x**2+x+9)/x))),x)

[Out]

2*log(log(((4*x - 5)*log(log((x**2 + x + 9)/x)) + 3)/log(log((x**2 + x + 9)/x))))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \, \log \left (\log \left (4 \, x \log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right ) - 5 \, \log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right ) + 3\right ) - \log \left (\log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right )\right )\right ) \]

[In]

integrate(((8*x^3+8*x^2+72*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x))^2-6*x^2+54)/((4*x^4-x^3+31*x^2-45*x)*log(
(x^2+x+9)/x)*log(log((x^2+x+9)/x))^2+(3*x^3+3*x^2+27*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x)))/log(((-5+4*x)*
log(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm="maxima")

[Out]

2*log(log(4*x*log(log(x^2 + x + 9) - log(x)) - 5*log(log(x^2 + x + 9) - log(x)) + 3) - log(log(log(x^2 + x + 9
) - log(x))))

Giac [F]

\[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=\int { \frac {2 \, {\left (4 \, {\left (x^{3} + x^{2} + 9 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )^{2} - 3 \, x^{2} + 27\right )}}{{\left ({\left (4 \, x^{4} - x^{3} + 31 \, x^{2} - 45 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )^{2} + 3 \, {\left (x^{3} + x^{2} + 9 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )\right )} \log \left (\frac {{\left (4 \, x - 5\right )} \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right ) + 3}{\log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )}\right )} \,d x } \]

[In]

integrate(((8*x^3+8*x^2+72*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x))^2-6*x^2+54)/((4*x^4-x^3+31*x^2-45*x)*log(
(x^2+x+9)/x)*log(log((x^2+x+9)/x))^2+(3*x^3+3*x^2+27*x)*log((x^2+x+9)/x)*log(log((x^2+x+9)/x)))/log(((-5+4*x)*
log(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm="giac")

[Out]

integrate(2*(4*(x^3 + x^2 + 9*x)*log((x^2 + x + 9)/x)*log(log((x^2 + x + 9)/x))^2 - 3*x^2 + 27)/(((4*x^4 - x^3
+ 31*x^2 - 45*x)*log((x^2 + x + 9)/x)*log(log((x^2 + x + 9)/x))^2 + 3*(x^3 + x^2 + 9*x)*log((x^2 + x + 9)/x)*
log(log((x^2 + x + 9)/x)))*log(((4*x - 5)*log(log((x^2 + x + 9)/x)) + 3)/log(log((x^2 + x + 9)/x)))), x)

Mupad [B] (verification not implemented)

Time = 16.74 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2\,\ln \left (\ln \left (\frac {\ln \left (\ln \left (\frac {x^2+x+9}{x}\right )\right )\,\left (4\,x-5\right )+3}{\ln \left (\ln \left (\frac {x^2+x+9}{x}\right )\right )}\right )\right ) \]

[In]

int((log((x + x^2 + 9)/x)*log(log((x + x^2 + 9)/x))^2*(72*x + 8*x^2 + 8*x^3) - 6*x^2 + 54)/(log((log(log((x +
x^2 + 9)/x))*(4*x - 5) + 3)/log(log((x + x^2 + 9)/x)))*(log((x + x^2 + 9)/x)*log(log((x + x^2 + 9)/x))*(27*x +
3*x^2 + 3*x^3) - log((x + x^2 + 9)/x)*log(log((x + x^2 + 9)/x))^2*(45*x - 31*x^2 + x^3 - 4*x^4))),x)

[Out]

2*log(log((log(log((x + x^2 + 9)/x))*(4*x - 5) + 3)/log(log((x + x^2 + 9)/x))))