Integrand size = 134, antiderivative size = 28 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\log (5)+\frac {3 \log (\log (x))}{x^2 \left (-3+x \left (-e^{e^5 x}+x\right )\right )^2} \] Output:
3*ln(ln(x))/(x*(x-exp(x*exp(5)))-3)^2/x^2+ln(5)
Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^2} \] Input:
Integrate[(9 + 3*E^(E^5*x)*x - 3*x^2 + (-18 + 18*x^2 + E^(E^5*x)*(-12*x - 6*E^5*x^2))*Log[x]*Log[Log[x]])/((27*x^3 - 27*x^5 + E^(3*E^5*x)*x^6 + 9*x^ 7 - x^9 + E^(2*E^5*x)*(9*x^5 - 3*x^7) + E^(E^5*x)*(27*x^4 - 18*x^6 + 3*x^8 ))*Log[x]),x]
Output:
(3*Log[Log[x]])/(x^2*(-3 - E^(E^5*x)*x + x^2)^2)
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.
\(\displaystyle \int \frac {-3 x^2+\left (18 x^2+e^{e^5 x} \left (-6 e^5 x^2-12 x\right )-18\right ) \log (x) \log (\log (x))+3 e^{e^5 x} x+9}{\left (-x^9+9 x^7+e^{3 e^5 x} x^6-27 x^5+27 x^3+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (3 x^8-18 x^6+27 x^4\right )\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 \left (-x^2-2 \left (e^{e^5 x+5} x^2-3 x^2+2 e^{e^5 x} x+3\right ) \log (x) \log (\log (x))+e^{e^5 x} x+3\right )}{x^3 \left (-x^2+e^{e^5 x} x+3\right )^3 \log (x)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {-x^2+e^{e^5 x} x-2 \left (e^{e^5 x+5} x^2-3 x^2+2 e^{e^5 x} x+3\right ) \log (x) \log (\log (x))+3}{x^3 \left (-x^2+e^{e^5 x} x+3\right )^3 \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 3 \int \left (\frac {2 \left (e^5 x^3-x^2-3 e^5 x-3\right ) \log (\log (x))}{x^3 \left (x^2-e^{e^5 x} x-3\right )^3}-\frac {2 e^5 x \log (x) \log (\log (x))+4 \log (x) \log (\log (x))-1}{x^3 \left (x^2-e^{e^5 x} x-3\right )^2 \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (-2 e^5 \int \frac {\log (\log (x))}{\left (-x^2+e^{e^5 x} x+3\right )^3}dx-6 e^5 \int \frac {\log (\log (x))}{x^2 \left (x^2-e^{e^5 x} x-3\right )^3}dx-2 \int \frac {\log (\log (x))}{x \left (x^2-e^{e^5 x} x-3\right )^3}dx-2 e^5 \int \frac {\log (\log (x))}{x^2 \left (x^2-e^{e^5 x} x-3\right )^2}dx+\int \frac {1}{x^3 \left (x^2-e^{e^5 x} x-3\right )^2 \log (x)}dx-6 \int \frac {\log (\log (x))}{x^3 \left (x^2-e^{e^5 x} x-3\right )^3}dx-4 \int \frac {\log (\log (x))}{x^3 \left (x^2-e^{e^5 x} x-3\right )^2}dx\right )\) |
Input:
Int[(9 + 3*E^(E^5*x)*x - 3*x^2 + (-18 + 18*x^2 + E^(E^5*x)*(-12*x - 6*E^5* x^2))*Log[x]*Log[Log[x]])/((27*x^3 - 27*x^5 + E^(3*E^5*x)*x^6 + 9*x^7 - x^ 9 + E^(2*E^5*x)*(9*x^5 - 3*x^7) + E^(E^5*x)*(27*x^4 - 18*x^6 + 3*x^8))*Log [x]),x]
Output:
$Aborted
Time = 302.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {3 \ln \left (\ln \left (x \right )\right )}{x^{2} \left (x^{2}-x \,{\mathrm e}^{x \,{\mathrm e}^{5}}-3\right )^{2}}\) | \(24\) |
parallelrisch | \(\frac {3 \ln \left (\ln \left (x \right )\right )}{x^{2} \left (x^{4}-2 \,{\mathrm e}^{x \,{\mathrm e}^{5}} x^{3}+{\mathrm e}^{2 x \,{\mathrm e}^{5}} x^{2}-6 x^{2}+6 x \,{\mathrm e}^{x \,{\mathrm e}^{5}}+9\right )}\) | \(50\) |
Input:
int((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*ln(x)*ln(ln(x))+3*x*ex p(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^2+( 3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/ln(x),x,method =_RETURNVERBOSE)
Output:
3/x^2/(x^2-x*exp(x*exp(5))-3)^2*ln(ln(x))
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{5} - 3 \, x^{3}\right )} e^{\left (x e^{5}\right )}} \] Input:
integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x ))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*ex p(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log(x ),x, algorithm="fricas")
Output:
3*log(log(x))/(x^6 + x^4*e^(2*x*e^5) - 6*x^4 + 9*x^2 - 2*(x^5 - 3*x^3)*e^( x*e^5))
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \log {\left (\log {\left (x \right )} \right )}}{x^{6} + x^{4} e^{2 x e^{5}} - 6 x^{4} + 9 x^{2} + \left (- 2 x^{5} + 6 x^{3}\right ) e^{x e^{5}}} \] Input:
integrate((((-6*x**2*exp(5)-12*x)*exp(x*exp(5))+18*x**2-18)*ln(x)*ln(ln(x) )+3*x*exp(x*exp(5))-3*x**2+9)/(x**6*exp(x*exp(5))**3+(-3*x**7+9*x**5)*exp( x*exp(5))**2+(3*x**8-18*x**6+27*x**4)*exp(x*exp(5))-x**9+9*x**7-27*x**5+27 *x**3)/ln(x),x)
Output:
3*log(log(x))/(x**6 + x**4*exp(2*x*exp(5)) - 6*x**4 + 9*x**2 + (-2*x**5 + 6*x**3)*exp(x*exp(5)))
Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{5} - 3 \, x^{3}\right )} e^{\left (x e^{5}\right )}} \] Input:
integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x ))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*ex p(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log(x ),x, algorithm="maxima")
Output:
3*log(log(x))/(x^6 + x^4*e^(2*x*e^5) - 6*x^4 + 9*x^2 - 2*(x^5 - 3*x^3)*e^( x*e^5))
Time = 0.18 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} - 2 \, x^{5} e^{\left (x e^{5}\right )} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 6 \, x^{3} e^{\left (x e^{5}\right )} + 9 \, x^{2}} \] Input:
integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x ))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*ex p(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log(x ),x, algorithm="giac")
Output:
3*log(log(x))/(x^6 - 2*x^5*e^(x*e^5) + x^4*e^(2*x*e^5) - 6*x^4 + 6*x^3*e^( x*e^5) + 9*x^2)
Time = 4.15 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3\,\ln \left (\ln \left (x\right )\right )}{x^2\,\left ({\mathrm {e}}^{x\,{\mathrm {e}}^5}\,\left (6\,x-2\,x^3\right )-6\,x^2+x^4+x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}+9\right )} \] Input:
int((3*x*exp(x*exp(5)) - 3*x^2 - log(log(x))*log(x)*(exp(x*exp(5))*(12*x + 6*x^2*exp(5)) - 18*x^2 + 18) + 9)/(log(x)*(exp(x*exp(5))*(27*x^4 - 18*x^6 + 3*x^8) + exp(2*x*exp(5))*(9*x^5 - 3*x^7) + 27*x^3 - 27*x^5 + 9*x^7 - x^ 9 + x^6*exp(3*x*exp(5)))),x)
Output:
(3*log(log(x)))/(x^2*(exp(x*exp(5))*(6*x - 2*x^3) - 6*x^2 + x^4 + x^2*exp( 2*x*exp(5)) + 9))
Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right )}{x^{2} \left (e^{2 e^{5} x} x^{2}-2 e^{e^{5} x} x^{3}+6 e^{e^{5} x} x +x^{4}-6 x^{2}+9\right )} \] Input:
int((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x))+3*x *exp(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^ 2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log(x),x)
Output:
(3*log(log(x)))/(x**2*(e**(2*e**5*x)*x**2 - 2*e**(e**5*x)*x**3 + 6*e**(e** 5*x)*x + x**4 - 6*x**2 + 9))