Integrand size = 131, antiderivative size = 40 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {x}{5 \left (\frac {\frac {x}{5}+\frac {\log (x)}{3}}{x}+\frac {e^{-6+2 x}}{x \log (\log (x))}\right )} \] Output:
1/5*x/((1/5*x+1/3*ln(x))/x+exp(-3+x)^2/x/ln(ln(x)))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 e^6 x^2 \log (\log (x))}{15 e^{2 x}+e^6 (3 x+5 \log (x)) \log (\log (x))} \] Input:
Integrate[(45*E^(-6 + 2*x)*x + E^(-6 + 2*x)*(90*x - 90*x^2)*Log[x]*Log[Log [x]] + ((-15*x + 9*x^2)*Log[x] + 30*x*Log[x]^2)*Log[Log[x]]^2)/(225*E^(-12 + 4*x)*Log[x] + (90*E^(-6 + 2*x)*x*Log[x] + 150*E^(-6 + 2*x)*Log[x]^2)*Lo g[Log[x]] + (9*x^2*Log[x] + 30*x*Log[x]^2 + 25*Log[x]^3)*Log[Log[x]]^2),x]
Output:
(3*E^6*x^2*Log[Log[x]])/(15*E^(2*x) + E^6*(3*x + 5*Log[x])*Log[Log[x]])
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.
| \(\displaystyle \int \frac {\left (\left (9 x^2-15 x\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))+e^{2 x-6} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+45 e^{2 x-6} x}{\left (9 x^2 \log (x)+25 \log ^3(x)+30 x \log ^2(x)\right ) \log ^2(\log (x))+\left (150 e^{2 x-6} \log ^2(x)+90 e^{2 x-6} x \log (x)\right ) \log (\log (x))+225 e^{4 x-12} \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{12} \left (\left (\left (9 x^2-15 x\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))+e^{2 x-6} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+45 e^{2 x-6} x\right )}{\log (x) \left (15 e^{2 x}+3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^{12} \int \frac {3 \left (-\left (\left (\left (5 x-3 x^2\right ) \log (x)-10 x \log ^2(x)\right ) \log ^2(\log (x))\right )+30 e^{2 x-6} \left (x-x^2\right ) \log (x) \log (\log (x))+15 e^{2 x-6} x\right )}{\log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 e^{12} \int \frac {-\left (\left (\left (5 x-3 x^2\right ) \log (x)-10 x \log ^2(x)\right ) \log ^2(\log (x))\right )+30 e^{2 x-6} \left (x-x^2\right ) \log (x) \log (\log (x))+15 e^{2 x-6} x}{\log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 3 e^{12} \int \left (\frac {x \log (\log (x)) \left (6 \log (x) \log (\log (x)) x^2+10 \log ^2(x) \log (\log (x)) x-3 \log (x) \log (\log (x)) x-3 x-5 \log (x)-5 \log (x) \log (\log (x))\right )}{\log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}-\frac {x (2 x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))-1)}{e^6 \log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 e^{12} \left (6 \int \frac {x^3 \log ^2(\log (x))}{\left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx-3 \int \frac {x^2 \log ^2(\log (x))}{\left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx+10 \int \frac {x^2 \log (x) \log ^2(\log (x))}{\left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx-3 \int \frac {x^2 \log (\log (x))}{\log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx-\frac {2 \int \frac {x^2 \log (\log (x))}{3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}}dx}{e^6}-5 \int \frac {x \log ^2(\log (x))}{\left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx-5 \int \frac {x \log (\log (x))}{\left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )^2}dx+\frac {\int \frac {x}{\log (x) \left (3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}\right )}dx}{e^6}+\frac {2 \int \frac {x \log (\log (x))}{3 e^6 x \log (\log (x))+5 e^6 \log (x) \log (\log (x))+15 e^{2 x}}dx}{e^6}\right )\) |
Input:
Int[(45*E^(-6 + 2*x)*x + E^(-6 + 2*x)*(90*x - 90*x^2)*Log[x]*Log[Log[x]] + ((-15*x + 9*x^2)*Log[x] + 30*x*Log[x]^2)*Log[Log[x]]^2)/(225*E^(-12 + 4*x )*Log[x] + (90*E^(-6 + 2*x)*x*Log[x] + 150*E^(-6 + 2*x)*Log[x]^2)*Log[Log[ x]] + (9*x^2*Log[x] + 30*x*Log[x]^2 + 25*Log[x]^3)*Log[Log[x]]^2),x]
Output:
$Aborted
Time = 3.72 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(\frac {3 \ln \left (\ln \left (x \right )\right ) x^{2}}{5 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x \ln \left (\ln \left (x \right )\right )+15 \,{\mathrm e}^{2 x -6}}\) | \(33\) |
| risch | \(\frac {3 x^{2}}{3 x +5 \ln \left (x \right )}-\frac {45 x^{2} {\mathrm e}^{2 x -6}}{\left (3 x +5 \ln \left (x \right )\right ) \left (5 \ln \left (x \right ) \ln \left (\ln \left (x \right )\right )+3 x \ln \left (\ln \left (x \right )\right )+15 \,{\mathrm e}^{2 x -6}\right )}\) | \(62\) |
Input:
int(((30*x*ln(x)^2+(9*x^2-15*x)*ln(x))*ln(ln(x))^2+(-90*x^2+90*x)*exp(-3+x )^2*ln(x)*ln(ln(x))+45*x*exp(-3+x)^2)/((25*ln(x)^3+30*x*ln(x)^2+9*x^2*ln(x ))*ln(ln(x))^2+(150*exp(-3+x)^2*ln(x)^2+90*x*exp(-3+x)^2*ln(x))*ln(ln(x))+ 225*exp(-3+x)^4*ln(x)),x,method=_RETURNVERBOSE)
Output:
3*x^2*ln(ln(x))/(5*ln(x)*ln(ln(x))+3*x*ln(ln(x))+15*exp(-3+x)^2)
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 \, x^{2} \log \left (\log \left (x\right )\right )}{{\left (3 \, x + 5 \, \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) + 15 \, e^{\left (2 \, x - 6\right )}} \] Input:
integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x )*exp(-3+x)^2*log(x)*log(log(x))+45*x*exp(-3+x)^2)/((25*log(x)^3+30*x*log( x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(-3+x)^2*log(x)^2+90*x*exp(-3+x)^ 2*log(x))*log(log(x))+225*exp(-3+x)^4*log(x)),x, algorithm="fricas")
Output:
3*x^2*log(log(x))/((3*x + 5*log(x))*log(log(x)) + 15*e^(2*x - 6))
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 x^{2} \log {\left (\log {\left (x \right )} \right )}}{3 x \log {\left (\log {\left (x \right )} \right )} + 15 e^{2 x - 6} + 5 \log {\left (x \right )} \log {\left (\log {\left (x \right )} \right )}} \] Input:
integrate(((30*x*ln(x)**2+(9*x**2-15*x)*ln(x))*ln(ln(x))**2+(-90*x**2+90*x )*exp(-3+x)**2*ln(x)*ln(ln(x))+45*x*exp(-3+x)**2)/((25*ln(x)**3+30*x*ln(x) **2+9*x**2*ln(x))*ln(ln(x))**2+(150*exp(-3+x)**2*ln(x)**2+90*x*exp(-3+x)** 2*ln(x))*ln(ln(x))+225*exp(-3+x)**4*ln(x)),x)
Output:
3*x**2*log(log(x))/(3*x*log(log(x)) + 15*exp(2*x - 6) + 5*log(x)*log(log(x )))
Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.88 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 \, x^{2} e^{6} \log \left (\log \left (x\right )\right )}{{\left (3 \, x e^{6} + 5 \, e^{6} \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) + 15 \, e^{\left (2 \, x\right )}} \] Input:
integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x )*exp(-3+x)^2*log(x)*log(log(x))+45*x*exp(-3+x)^2)/((25*log(x)^3+30*x*log( x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(-3+x)^2*log(x)^2+90*x*exp(-3+x)^ 2*log(x))*log(log(x))+225*exp(-3+x)^4*log(x)),x, algorithm="maxima")
Output:
3*x^2*e^6*log(log(x))/((3*x*e^6 + 5*e^6*log(x))*log(log(x)) + 15*e^(2*x))
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 \, x^{2} e^{6} \log \left (\log \left (x\right )\right )}{3 \, x e^{6} \log \left (\log \left (x\right )\right ) + 5 \, e^{6} \log \left (x\right ) \log \left (\log \left (x\right )\right ) + 15 \, e^{\left (2 \, x\right )}} \] Input:
integrate(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x )*exp(-3+x)^2*log(x)*log(log(x))+45*x*exp(-3+x)^2)/((25*log(x)^3+30*x*log( x)^2+9*x^2*log(x))*log(log(x))^2+(150*exp(-3+x)^2*log(x)^2+90*x*exp(-3+x)^ 2*log(x))*log(log(x))+225*exp(-3+x)^4*log(x)),x, algorithm="giac")
Output:
3*x^2*e^6*log(log(x))/(3*x*e^6*log(log(x)) + 5*e^6*log(x)*log(log(x)) + 15 *e^(2*x))
Time = 1.86 (sec) , antiderivative size = 280, normalized size of antiderivative = 7.00 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=2\,x+\frac {50}{9\,\left (x+\frac {5}{3}\right )}-\frac {\frac {3\,x^2\,\left (3\,x-5\right )}{3\,x+5}+\frac {30\,x^2\,\ln \left (x\right )}{3\,x+5}}{3\,x+5\,\ln \left (x\right )}-\frac {45\,\left (25\,x^3\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^3+30\,x^4\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^2-75\,x^3\,{\mathrm {e}}^{4\,x-12}\,{\ln \left (x\right )}^2-45\,x^4\,{\mathrm {e}}^{4\,x-12}\,{\ln \left (x\right )}^2+150\,x^4\,{\mathrm {e}}^{4\,x-12}\,{\ln \left (x\right )}^3+90\,x^5\,{\mathrm {e}}^{4\,x-12}\,{\ln \left (x\right )}^2+9\,x^5\,{\mathrm {e}}^{2\,x-6}\,\ln \left (x\right )\right )}{\left (3\,x+5\,\ln \left (x\right )\right )\,\left (15\,{\mathrm {e}}^{2\,x-6}+\ln \left (\ln \left (x\right )\right )\,\left (3\,x+5\,\ln \left (x\right )\right )\right )\,\left (25\,x\,{\ln \left (x\right )}^3+9\,x^3\,\ln \left (x\right )+30\,x^2\,{\ln \left (x\right )}^2-45\,x^2\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^2+150\,x^2\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^3+90\,x^3\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^2-75\,x\,{\mathrm {e}}^{2\,x-6}\,{\ln \left (x\right )}^2\right )} \] Input:
int((log(log(x))^2*(30*x*log(x)^2 - log(x)*(15*x - 9*x^2)) + 45*x*exp(2*x - 6) + log(log(x))*exp(2*x - 6)*log(x)*(90*x - 90*x^2))/(log(log(x))^2*(30 *x*log(x)^2 + 9*x^2*log(x) + 25*log(x)^3) + log(log(x))*(150*exp(2*x - 6)* log(x)^2 + 90*x*exp(2*x - 6)*log(x)) + 225*exp(4*x - 12)*log(x)),x)
Output:
2*x + 50/(9*(x + 5/3)) - ((3*x^2*(3*x - 5))/(3*x + 5) + (30*x^2*log(x))/(3 *x + 5))/(3*x + 5*log(x)) - (45*(25*x^3*exp(2*x - 6)*log(x)^3 + 30*x^4*exp (2*x - 6)*log(x)^2 - 75*x^3*exp(4*x - 12)*log(x)^2 - 45*x^4*exp(4*x - 12)* log(x)^2 + 150*x^4*exp(4*x - 12)*log(x)^3 + 90*x^5*exp(4*x - 12)*log(x)^2 + 9*x^5*exp(2*x - 6)*log(x)))/((3*x + 5*log(x))*(15*exp(2*x - 6) + log(log (x))*(3*x + 5*log(x)))*(25*x*log(x)^3 + 9*x^3*log(x) + 30*x^2*log(x)^2 - 4 5*x^2*exp(2*x - 6)*log(x)^2 + 150*x^2*exp(2*x - 6)*log(x)^3 + 90*x^3*exp(2 *x - 6)*log(x)^2 - 75*x*exp(2*x - 6)*log(x)^2))
Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {45 e^{-6+2 x} x+e^{-6+2 x} \left (90 x-90 x^2\right ) \log (x) \log (\log (x))+\left (\left (-15 x+9 x^2\right ) \log (x)+30 x \log ^2(x)\right ) \log ^2(\log (x))}{225 e^{-12+4 x} \log (x)+\left (90 e^{-6+2 x} x \log (x)+150 e^{-6+2 x} \log ^2(x)\right ) \log (\log (x))+\left (9 x^2 \log (x)+30 x \log ^2(x)+25 \log ^3(x)\right ) \log ^2(\log (x))} \, dx=\frac {3 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) e^{6} x^{2}}{15 e^{2 x}+5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (x \right ) e^{6}+3 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) e^{6} x} \] Input:
int(((30*x*log(x)^2+(9*x^2-15*x)*log(x))*log(log(x))^2+(-90*x^2+90*x)*exp( -3+x)^2*log(x)*log(log(x))+45*x*exp(-3+x)^2)/((25*log(x)^3+30*x*log(x)^2+9 *x^2*log(x))*log(log(x))^2+(150*exp(-3+x)^2*log(x)^2+90*x*exp(-3+x)^2*log( x))*log(log(x))+225*exp(-3+x)^4*log(x)),x)
Output:
(3*log(log(x))*e**6*x**2)/(15*e**(2*x) + 5*log(log(x))*log(x)*e**6 + 3*log (log(x))*e**6*x)