Integrand size = 172, antiderivative size = 23 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=\log \left (\log ^2\left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )\right ) \]
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \log \left (\log \left (-5+4 x+\frac {3}{\log \left (\log \left (1+\frac {9}{x}+x\right )\right )}\right )\right ) \]
Integrate[(54 - 6*x^2 + (72*x + 8*x^2 + 8*x^3)*Log[(9 + x + x^2)/x]*Log[Lo g[(9 + x + x^2)/x]]^2)/(((27*x + 3*x^2 + 3*x^3)*Log[(9 + x + x^2)/x]*Log[L og[(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]
Time = 13.62 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {7292, 7279, 7239, 7235}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {-6 x^2+\left (8 x^3+8 x^2+72 x\right ) \log \left (\frac {x^2+x+9}{x}\right ) \log ^2\left (\log \left (\frac {x^2+x+9}{x}\right )\right )+54}{\left (\left (3 x^3+3 x^2+27 x\right ) \log \left (\frac {x^2+x+9}{x}\right ) \log \left (\log \left (\frac {x^2+x+9}{x}\right )\right )+\left (4 x^4-x^3+31 x^2-45 x\right ) \log \left (\frac {x^2+x+9}{x}\right ) \log ^2\left (\log \left (\frac {x^2+x+9}{x}\right )\right )\right ) \log \left (\frac {(4 x-5) \log \left (\log \left (\frac {x^2+x+9}{x}\right )\right )+3}{\log \left (\log \left (\frac {x^2+x+9}{x}\right )\right )}\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-6 x^2+\left (8 x^3+8 x^2+72 x\right ) \log \left (\frac {x^2+x+9}{x}\right ) \log ^2\left (\log \left (\frac {x^2+x+9}{x}\right )\right )+54}{x \left (x^2+x+9\right ) \log \left (x+\frac {9}{x}+1\right ) \log \left (\log \left (x+\frac {9}{x}+1\right )\right ) \left (4 x \log \left (\log \left (x+\frac {9}{x}+1\right )\right )-5 \log \left (\log \left (x+\frac {9}{x}+1\right )\right )+3\right ) \log \left (\frac {(4 x-5) \log \left (\log \left (\frac {x^2+x+9}{x}\right )\right )+3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}\right )}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {2 \left (4 x^3 \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )-3 x^2+4 x^2 \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )+36 x \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )+27\right )}{9 x \log \left (x+\frac {9}{x}+1\right ) \log \left (\log \left (x+\frac {9}{x}+1\right )\right ) \left (4 x \log \left (\log \left (x+\frac {9}{x}+1\right )\right )-5 \log \left (\log \left (x+\frac {9}{x}+1\right )\right )+3\right ) \log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )}-\frac {2 (x+1) \left (4 x^3 \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )-3 x^2+4 x^2 \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )+36 x \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )+27\right )}{9 \left (x^2+x+9\right ) \log \left (x+\frac {9}{x}+1\right ) \log \left (\log \left (x+\frac {9}{x}+1\right )\right ) \left (4 x \log \left (\log \left (x+\frac {9}{x}+1\right )\right )-5 \log \left (\log \left (x+\frac {9}{x}+1\right )\right )+3\right ) \log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 x \left (x^2+x+9\right ) \log \left (x+\frac {9}{x}+1\right ) \log ^2\left (\log \left (x+\frac {9}{x}+1\right )\right )-6 \left (x^2-9\right )}{x \left (x^2+x+9\right ) \log \left (x+\frac {9}{x}+1\right ) \log \left (\log \left (x+\frac {9}{x}+1\right )\right ) \left ((4 x-5) \log \left (\log \left (x+\frac {9}{x}+1\right )\right )+3\right ) \log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle 2 \log \left (\log \left (4 x+\frac {3}{\log \left (\log \left (x+\frac {9}{x}+1\right )\right )}-5\right )\right )\) |
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[L og[(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]
3.14.30.3.1 Defintions of rubi rules used
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
\[\int \frac {\left (8 x^{3}+8 x^{2}+72 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) {\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}^{2}-6 x^{2}+54}{\left (\left (4 x^{4}-x^{3}+31 x^{2}-45 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) {\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}^{2}+\left (3 x^{3}+3 x^{2}+27 x \right ) \ln \left (\frac {x^{2}+x +9}{x}\right ) \ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )\right ) \ln \left (\frac {\left (-5+4 x \right ) \ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )+3}{\ln \left (\ln \left (\frac {x^{2}+x +9}{x}\right )\right )}\right )}d x\]
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)
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)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \, \log \left (\log \left (\frac {{\left (4 \, x - 5\right )} \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right ) + 3}{\log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )}\right )\right ) \]
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)*lo g(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm=\
Time = 3.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \log {\left (\log {\left (\frac {\left (4 x - 5\right ) \log {\left (\log {\left (\frac {x^{2} + x + 9}{x} \right )} \right )} + 3}{\log {\left (\log {\left (\frac {x^{2} + x + 9}{x} \right )} \right )}} \right )} \right )} \]
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)
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 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2 \, \log \left (\log \left (4 \, x \log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right ) - 5 \, \log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right ) + 3\right ) - \log \left (\log \left (\log \left (x^{2} + x + 9\right ) - \log \left (x\right )\right )\right )\right ) \]
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)*lo g(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm=\
2*log(log(4*x*log(log(x^2 + x + 9) - log(x)) - 5*log(log(x^2 + x + 9) - lo g(x)) + 3) - log(log(log(x^2 + x + 9) - log(x))))
\[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=\int { \frac {2 \, {\left (4 \, {\left (x^{3} + x^{2} + 9 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )^{2} - 3 \, x^{2} + 27\right )}}{{\left ({\left (4 \, x^{4} - x^{3} + 31 \, x^{2} - 45 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )^{2} + 3 \, {\left (x^{3} + x^{2} + 9 \, x\right )} \log \left (\frac {x^{2} + x + 9}{x}\right ) \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )\right )} \log \left (\frac {{\left (4 \, x - 5\right )} \log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right ) + 3}{\log \left (\log \left (\frac {x^{2} + x + 9}{x}\right )\right )}\right )} \,d x } \]
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)*lo g(log((x^2+x+9)/x))+3)/log(log((x^2+x+9)/x))),x, algorithm=\
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)*lo g(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)
Time = 16.74 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {54-6 x^2+\left (72 x+8 x^2+8 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\left (\left (27 x+3 x^2+3 x^3\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )+\left (-45 x+31 x^2-x^3+4 x^4\right ) \log \left (\frac {9+x+x^2}{x}\right ) \log ^2\left (\log \left (\frac {9+x+x^2}{x}\right )\right )\right ) \log \left (\frac {3+(-5+4 x) \log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}{\log \left (\log \left (\frac {9+x+x^2}{x}\right )\right )}\right )} \, dx=2\,\ln \left (\ln \left (\frac {\ln \left (\ln \left (\frac {x^2+x+9}{x}\right )\right )\,\left (4\,x-5\right )+3}{\ln \left (\ln \left (\frac {x^2+x+9}{x}\right )\right )}\right )\right ) \]
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 - 3 1*x^2 + x^3 - 4*x^4))),x)