Integrand size = 122, antiderivative size = 25 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {x}{-3-e^{\frac {4 x}{\log (\log (x))}}-(-1+x)^2} \]
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {x}{4+e^{\frac {4 x}{\log (\log (x))}}-2 x+x^2} \]
Integrate[((-4 + x^2)*Log[x]*Log[Log[x]]^2 + E^((4*x)/Log[Log[x]])*(-4*x + 4*x*Log[x]*Log[Log[x]] - Log[x]*Log[Log[x]]^2))/(E^((8*x)/Log[Log[x]])*Lo g[x]*Log[Log[x]]^2 + E^((4*x)/Log[Log[x]])*(8 - 4*x + 2*x^2)*Log[x]*Log[Lo g[x]]^2 + (16 - 16*x + 12*x^2 - 4*x^3 + x^4)*Log[x]*Log[Log[x]]^2),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.
| \(\displaystyle \int \frac {\left (x^2-4\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x-\log (x) \log ^2(\log (x))+4 x \log (x) \log (\log (x))\right )}{\left (2 x^2-4 x+8\right ) e^{\frac {4 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+\left (x^4-4 x^3+12 x^2-16 x+16\right ) \log (x) \log ^2(\log (x))+e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (x^2-4\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x-\log (x) \log ^2(\log (x))+4 x \log (x) \log (\log (x))\right )}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (x) \log ^2(\log (x))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {-4 x-\log (x) \log ^2(\log (x))+4 x \log (x) \log (\log (x))}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right ) \log (x) \log ^2(\log (x))}-\frac {2 x \left (-2 x^2+2 x^2 \log (x) \log (\log (x))+4 x-x \log (x) \log ^2(\log (x))+\log (x) \log ^2(\log (x))-4 x \log (x) \log (\log (x))+8 \log (x) \log (\log (x))-8\right )}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (x) \log ^2(\log (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \int \frac {x}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (x) \log ^2(\log (x))}dx-8 \int \frac {x^2}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (x) \log ^2(\log (x))}dx-4 \int \frac {x}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right ) \log (x) \log ^2(\log (x))}dx-2 \int \frac {x}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2}dx+2 \int \frac {x^2}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2}dx-\int \frac {1}{x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4}dx-16 \int \frac {x}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (\log (x))}dx+8 \int \frac {x^2}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (\log (x))}dx+4 \int \frac {x}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right ) \log (\log (x))}dx+4 \int \frac {x^3}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (x) \log ^2(\log (x))}dx-4 \int \frac {x^3}{\left (x^2-2 x+e^{\frac {4 x}{\log (\log (x))}}+4\right )^2 \log (\log (x))}dx\) |
Int[((-4 + x^2)*Log[x]*Log[Log[x]]^2 + E^((4*x)/Log[Log[x]])*(-4*x + 4*x*L og[x]*Log[Log[x]] - Log[x]*Log[Log[x]]^2))/(E^((8*x)/Log[Log[x]])*Log[x]*L og[Log[x]]^2 + E^((4*x)/Log[Log[x]])*(8 - 4*x + 2*x^2)*Log[x]*Log[Log[x]]^ 2 + (16 - 16*x + 12*x^2 - 4*x^3 + x^4)*Log[x]*Log[Log[x]]^2),x]
3.24.20.3.1 Defintions of rubi rules used
Time = 37.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(-\frac {x}{{\mathrm e}^{\frac {4 x}{\ln \left (\ln \left (x \right )\right )}}+x^{2}-2 x +4}\) | \(23\) |
| parallelrisch | \(-\frac {x}{{\mathrm e}^{\frac {4 x}{\ln \left (\ln \left (x \right )\right )}}+x^{2}-2 x +4}\) | \(25\) |
int(((-ln(x)*ln(ln(x))^2+4*x*ln(x)*ln(ln(x))-4*x)*exp(2*x/ln(ln(x)))^2+(x^ 2-4)*ln(x)*ln(ln(x))^2)/(ln(x)*ln(ln(x))^2*exp(2*x/ln(ln(x)))^4+(2*x^2-4*x +8)*ln(x)*ln(ln(x))^2*exp(2*x/ln(ln(x)))^2+(x^4-4*x^3+12*x^2-16*x+16)*ln(x )*ln(ln(x))^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {x}{x^{2} - 2 \, x + e^{\left (\frac {4 \, x}{\log \left (\log \left (x\right )\right )}\right )} + 4} \]
integrate(((-log(x)*log(log(x))^2+4*x*log(x)*log(log(x))-4*x)*exp(2*x/log( log(x)))^2+(x^2-4)*log(x)*log(log(x))^2)/(log(x)*log(log(x))^2*exp(2*x/log (log(x)))^4+(2*x^2-4*x+8)*log(x)*log(log(x))^2*exp(2*x/log(log(x)))^2+(x^4 -4*x^3+12*x^2-16*x+16)*log(x)*log(log(x))^2),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=- \frac {x}{x^{2} - 2 x + e^{\frac {4 x}{\log {\left (\log {\left (x \right )} \right )}}} + 4} \]
integrate(((-ln(x)*ln(ln(x))**2+4*x*ln(x)*ln(ln(x))-4*x)*exp(2*x/ln(ln(x)) )**2+(x**2-4)*ln(x)*ln(ln(x))**2)/(ln(x)*ln(ln(x))**2*exp(2*x/ln(ln(x)))** 4+(2*x**2-4*x+8)*ln(x)*ln(ln(x))**2*exp(2*x/ln(ln(x)))**2+(x**4-4*x**3+12* x**2-16*x+16)*ln(x)*ln(ln(x))**2),x)
Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {x}{x^{2} - 2 \, x + e^{\left (\frac {4 \, x}{\log \left (\log \left (x\right )\right )}\right )} + 4} \]
integrate(((-log(x)*log(log(x))^2+4*x*log(x)*log(log(x))-4*x)*exp(2*x/log( log(x)))^2+(x^2-4)*log(x)*log(log(x))^2)/(log(x)*log(log(x))^2*exp(2*x/log (log(x)))^4+(2*x^2-4*x+8)*log(x)*log(log(x))^2*exp(2*x/log(log(x)))^2+(x^4 -4*x^3+12*x^2-16*x+16)*log(x)*log(log(x))^2),x, algorithm=\
\[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=\int { \frac {{\left (x^{2} - 4\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + {\left (4 \, x \log \left (x\right ) \log \left (\log \left (x\right )\right ) - \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - 4 \, x\right )} e^{\left (\frac {4 \, x}{\log \left (\log \left (x\right )\right )}\right )}}{2 \, {\left (x^{2} - 2 \, x + 4\right )} e^{\left (\frac {4 \, x}{\log \left (\log \left (x\right )\right )}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 16 \, x + 16\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + e^{\left (\frac {8 \, x}{\log \left (\log \left (x\right )\right )}\right )} \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2}} \,d x } \]
integrate(((-log(x)*log(log(x))^2+4*x*log(x)*log(log(x))-4*x)*exp(2*x/log( log(x)))^2+(x^2-4)*log(x)*log(log(x))^2)/(log(x)*log(log(x))^2*exp(2*x/log (log(x)))^4+(2*x^2-4*x+8)*log(x)*log(log(x))^2*exp(2*x/log(log(x)))^2+(x^4 -4*x^3+12*x^2-16*x+16)*log(x)*log(log(x))^2),x, algorithm=\
Time = 13.71 (sec) , antiderivative size = 209, normalized size of antiderivative = 8.36 \[ \int \frac {\left (-4+x^2\right ) \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (-4 x+4 x \log (x) \log (\log (x))-\log (x) \log ^2(\log (x))\right )}{e^{\frac {8 x}{\log (\log (x))}} \log (x) \log ^2(\log (x))+e^{\frac {4 x}{\log (\log (x))}} \left (8-4 x+2 x^2\right ) \log (x) \log ^2(\log (x))+\left (16-16 x+12 x^2-4 x^3+x^4\right ) \log (x) \log ^2(\log (x))} \, dx=-\frac {x\,\left ({\ln \left (\ln \left (x\right )\right )}^4\,{\ln \left (x\right )}^2+8\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2-8\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\right )-x^2\,\left ({\ln \left (\ln \left (x\right )\right )}^4\,{\ln \left (x\right )}^2+4\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2-4\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\right )+x^3\,\left (2\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2-2\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\right )}{\left ({\mathrm {e}}^{\frac {4\,x}{\ln \left (\ln \left (x\right )\right )}}-2\,x+x^2+4\right )\,\left (2\,x^2\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2-2\,x^2\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )-x\,{\ln \left (\ln \left (x\right )\right )}^4\,{\ln \left (x\right )}^2-4\,x\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2+4\,x\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )+{\ln \left (\ln \left (x\right )\right )}^4\,{\ln \left (x\right )}^2+8\,{\ln \left (\ln \left (x\right )\right )}^3\,{\ln \left (x\right )}^2-8\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )\right )} \]
int(-(exp((4*x)/log(log(x)))*(4*x + log(log(x))^2*log(x) - 4*x*log(log(x)) *log(x)) - log(log(x))^2*log(x)*(x^2 - 4))/(log(log(x))^2*log(x)*(12*x^2 - 16*x - 4*x^3 + x^4 + 16) + log(log(x))^2*exp((8*x)/log(log(x)))*log(x) + log(log(x))^2*exp((4*x)/log(log(x)))*log(x)*(2*x^2 - 4*x + 8)),x)
-(x*(8*log(log(x))^3*log(x)^2 + log(log(x))^4*log(x)^2 - 8*log(log(x))^2*l og(x)) - x^2*(4*log(log(x))^3*log(x)^2 + log(log(x))^4*log(x)^2 - 4*log(lo g(x))^2*log(x)) + x^3*(2*log(log(x))^3*log(x)^2 - 2*log(log(x))^2*log(x))) /((exp((4*x)/log(log(x))) - 2*x + x^2 + 4)*(8*log(log(x))^3*log(x)^2 + log (log(x))^4*log(x)^2 - 8*log(log(x))^2*log(x) + 2*x^2*log(log(x))^3*log(x)^ 2 + 4*x*log(log(x))^2*log(x) - 2*x^2*log(log(x))^2*log(x) - 4*x*log(log(x) )^3*log(x)^2 - x*log(log(x))^4*log(x)^2))