Integrand size = 108, antiderivative size = 25 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x+\frac {1-\frac {-2+x}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}}{x} \]
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\frac {1}{x}+x+\frac {2-x}{x \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \]
Integrate[(-2*x + x^2 - 2*Log[E^x/2]*Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]] + (-1 + x^2)*Log[E^x/2]*Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]]^2)/(x^2*Log[E ^x/2]*Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]]^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.
\(\displaystyle \int \frac {x^2+\left (x^2-1\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )-2 x-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2-1}{x^2}-\frac {2}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}+\frac {x-2}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{x^2 \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}dx-2 \int \frac {1}{x \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}dx+x+\frac {1}{x}-\frac {1}{\log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}\) |
Int[(-2*x + x^2 - 2*Log[E^x/2]*Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]] + (-1 + x^2)*Log[E^x/2]*Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]]^2)/(x^2*Log[E^x/2]* Log[Log[E^x/2]]*Log[Log[Log[E^x/2]]]^2),x]
3.29.16.3.1 Defintions of rubi rules used
Time = 4.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {x^{2}+1}{x}-\frac {-2+x}{x \ln \left (\ln \left (-\ln \left (2\right )+\ln \left ({\mathrm e}^{x}\right )\right )\right )}\) | \(31\) |
parallelrisch | \(-\frac {-12-6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right ) x^{2}+6 x -6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right )}{6 \ln \left (\ln \left (\ln \left (\frac {{\mathrm e}^{x}}{2}\right )\right )\right ) x}\) | \(41\) |
int(((x^2-1)*ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln(ln(ln(1/2*exp(x))))^2-2* ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln(ln(ln(1/2*exp(x))))+x^2-2*x)/x^2/ln(1 /2*exp(x))/ln(ln(1/2*exp(x)))/ln(ln(ln(1/2*exp(x))))^2,x,method=_RETURNVER BOSE)
Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\frac {{\left (x^{2} + 1\right )} \log \left (\log \left (x - \log \left (2\right )\right )\right ) - x + 2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]
integrate(((x^2-1)*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*ex p(x))))^2-2*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*exp(x)))) +x^2-2*x)/x^2/log(1/2*exp(x))/log(log(1/2*exp(x)))/log(log(log(1/2*exp(x)) ))^2,x, algorithm=\
Timed out. \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=\text {Timed out} \]
integrate(((x**2-1)*ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln(ln(ln(1/2*exp(x)) ))**2-2*ln(1/2*exp(x))*ln(ln(1/2*exp(x)))*ln(ln(ln(1/2*exp(x))))+x**2-2*x) /x**2/ln(1/2*exp(x))/ln(ln(1/2*exp(x)))/ln(ln(ln(1/2*exp(x))))**2,x)
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x + \frac {1}{x} - \frac {1}{\log \left (\log \left (\log \left (\frac {1}{2} \, e^{x}\right )\right )\right )} + \frac {2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]
integrate(((x^2-1)*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*ex p(x))))^2-2*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*exp(x)))) +x^2-2*x)/x^2/log(1/2*exp(x))/log(log(1/2*exp(x)))/log(log(log(1/2*exp(x)) ))^2,x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x + \frac {1}{x} - \frac {x - 2}{x \log \left (\log \left (x - \log \left (2\right )\right )\right )} \]
integrate(((x^2-1)*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*ex p(x))))^2-2*log(1/2*exp(x))*log(log(1/2*exp(x)))*log(log(log(1/2*exp(x)))) +x^2-2*x)/x^2/log(1/2*exp(x))/log(log(1/2*exp(x)))/log(log(log(1/2*exp(x)) ))^2,x, algorithm=\
Time = 10.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {-2 x+x^2-2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log \left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )+\left (-1+x^2\right ) \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )}{x^2 \log \left (\frac {e^x}{2}\right ) \log \left (\log \left (\frac {e^x}{2}\right )\right ) \log ^2\left (\log \left (\log \left (\frac {e^x}{2}\right )\right )\right )} \, dx=x+\frac {2}{x\,\ln \left (\ln \left (x-\ln \left (2\right )\right )\right )}+\frac {1}{x}-\frac {1}{\ln \left (\ln \left (x-\ln \left (2\right )\right )\right )} \]