Integrand size = 161, antiderivative size = 26 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=4+\frac {3 e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \]
Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=\frac {3 e^x}{x^2 \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \]
Integrate[(6*E^x + (-12*E^x - 3*E^x*Log[x^2])*Log[4 + Log[x^2]] + (E^x*(-2 4*x + 12*x^2) + E^x*(-6*x + 3*x^2)*Log[x^2] + (E^x*(-24 + 12*x) + E^x*(-6 + 3*x)*Log[x^2])*Log[4 + Log[x^2]])*Log[x/(x + Log[4 + Log[x^2]])])/((4*x^ 4 + x^4*Log[x^2] + (4*x^3 + x^3*Log[x^2])*Log[4 + Log[x^2]])*Log[x/(x + Lo g[4 + Log[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.
| \(\displaystyle \int \frac {\left (-3 e^x \log \left (x^2\right )-12 e^x\right ) \log \left (\log \left (x^2\right )+4\right )+\left (e^x \left (12 x^2-24 x\right )+e^x \left (3 x^2-6 x\right ) \log \left (x^2\right )+\left (e^x (3 x-6) \log \left (x^2\right )+e^x (12 x-24)\right ) \log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )+6 e^x}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )+4\right )\right ) \log ^2\left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-3 e^x \log \left (x^2\right )-12 e^x\right ) \log \left (\log \left (x^2\right )+4\right )+\left (e^x \left (12 x^2-24 x\right )+e^x \left (3 x^2-6 x\right ) \log \left (x^2\right )+\left (e^x (3 x-6) \log \left (x^2\right )+e^x (12 x-24)\right ) \log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )+6 e^x}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log ^2\left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 e^x \log \left (\log \left (x^2\right )+4\right ) \log \left (x^2\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}+\frac {3 e^x \log \left (x^2\right )}{x \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {6 e^x \log \left (x^2\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}+\frac {12 e^x \log \left (\log \left (x^2\right )+4\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}+\frac {12 e^x}{x \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {24 e^x}{x^2 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {3 e^x \log \left (\log \left (x^2\right )+4\right ) \log \left (x^2\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log ^2\left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {12 e^x \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log ^2\left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}+\frac {6 e^x}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log ^2\left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {6 e^x \log \left (\log \left (x^2\right )+4\right ) \log \left (x^2\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}-\frac {24 e^x \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (\log \left (\log \left (x^2\right )+4\right )+x\right ) \log \left (\frac {x}{\log \left (\log \left (x^2\right )+4\right )+x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -24 \int \frac {e^x}{x^2 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx+12 \int \frac {e^x}{x \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx-6 \int \frac {e^x \log \left (x^2\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx+3 \int \frac {e^x \log \left (x^2\right )}{x \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx+12 \int \frac {e^x \log \left (\log \left (x^2\right )+4\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx+3 \int \frac {e^x \log \left (x^2\right ) \log \left (\log \left (x^2\right )+4\right )}{x^2 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx+6 \int \frac {e^x}{x^3 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log ^2\left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx-12 \int \frac {e^x \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log ^2\left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx-3 \int \frac {e^x \log \left (x^2\right ) \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log ^2\left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx-24 \int \frac {e^x \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx-6 \int \frac {e^x \log \left (x^2\right ) \log \left (\log \left (x^2\right )+4\right )}{x^3 \left (\log \left (x^2\right )+4\right ) \left (x+\log \left (\log \left (x^2\right )+4\right )\right ) \log \left (\frac {x}{x+\log \left (\log \left (x^2\right )+4\right )}\right )}dx\) |
Int[(6*E^x + (-12*E^x - 3*E^x*Log[x^2])*Log[4 + Log[x^2]] + (E^x*(-24*x + 12*x^2) + E^x*(-6*x + 3*x^2)*Log[x^2] + (E^x*(-24 + 12*x) + E^x*(-6 + 3*x) *Log[x^2])*Log[4 + Log[x^2]])*Log[x/(x + Log[4 + Log[x^2]])])/((4*x^4 + x^ 4*Log[x^2] + (4*x^3 + x^3*Log[x^2])*Log[4 + Log[x^2]])*Log[x/(x + Log[4 + Log[x^2]])]^2),x]
3.8.93.3.1 Defintions of rubi rules used
Time = 154.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
| method | result | size |
| parallelrisch | \(\frac {3 \,{\mathrm e}^{x}}{x^{2} \ln \left (\frac {x}{\ln \left (4+\ln \left (x^{2}\right )\right )+x}\right )}\) | \(24\) |
| risch | \(\frac {6 i {\mathrm e}^{x}}{x^{2} \left (\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right )-\pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right ) {\operatorname {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i x}{\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x}\right )}^{3}+2 i \ln \left (x \right )-2 i \ln \left (\ln \left (4+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )+x \right )\right )}\) | \(346\) |
int(((((-6+3*x)*exp(x)*ln(x^2)+(12*x-24)*exp(x))*ln(4+ln(x^2))+(3*x^2-6*x) *exp(x)*ln(x^2)+(12*x^2-24*x)*exp(x))*ln(x/(ln(4+ln(x^2))+x))+(-3*exp(x)*l n(x^2)-12*exp(x))*ln(4+ln(x^2))+6*exp(x))/((x^3*ln(x^2)+4*x^3)*ln(4+ln(x^2 ))+x^4*ln(x^2)+4*x^4)/ln(x/(ln(4+ln(x^2))+x))^2,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=\frac {3 \, e^{x}}{x^{2} \log \left (\frac {x}{x + \log \left (\log \left (x^{2}\right ) + 4\right )}\right )} \]
integrate(((((-6+3*x)*exp(x)*log(x^2)+(12*x-24)*exp(x))*log(4+log(x^2))+(3 *x^2-6*x)*exp(x)*log(x^2)+(12*x^2-24*x)*exp(x))*log(x/(log(4+log(x^2))+x)) +(-3*exp(x)*log(x^2)-12*exp(x))*log(4+log(x^2))+6*exp(x))/((x^3*log(x^2)+4 *x^3)*log(4+log(x^2))+x^4*log(x^2)+4*x^4)/log(x/(log(4+log(x^2))+x))^2,x, algorithm=\
Time = 1.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=\frac {3 e^{x}}{x^{2} \log {\left (\frac {x}{x + \log {\left (\log {\left (x^{2} \right )} + 4 \right )}} \right )}} \]
integrate(((((-6+3*x)*exp(x)*ln(x**2)+(12*x-24)*exp(x))*ln(4+ln(x**2))+(3* x**2-6*x)*exp(x)*ln(x**2)+(12*x**2-24*x)*exp(x))*ln(x/(ln(4+ln(x**2))+x))+ (-3*exp(x)*ln(x**2)-12*exp(x))*ln(4+ln(x**2))+6*exp(x))/((x**3*ln(x**2)+4* x**3)*ln(4+ln(x**2))+x**4*ln(x**2)+4*x**4)/ln(x/(ln(4+ln(x**2))+x))**2,x)
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=-\frac {3 \, e^{x}}{x^{2} \log \left (x + \log \left (2\right ) + \log \left (\log \left (x\right ) + 2\right )\right ) - x^{2} \log \left (x\right )} \]
integrate(((((-6+3*x)*exp(x)*log(x^2)+(12*x-24)*exp(x))*log(4+log(x^2))+(3 *x^2-6*x)*exp(x)*log(x^2)+(12*x^2-24*x)*exp(x))*log(x/(log(4+log(x^2))+x)) +(-3*exp(x)*log(x^2)-12*exp(x))*log(4+log(x^2))+6*exp(x))/((x^3*log(x^2)+4 *x^3)*log(4+log(x^2))+x^4*log(x^2)+4*x^4)/log(x/(log(4+log(x^2))+x))^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (25) = 50\).
Time = 0.80 (sec) , antiderivative size = 288, normalized size of antiderivative = 11.08 \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=-\frac {3 \, {\left (e^{x} \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) + 2 \, e^{x} \log \left (x^{2}\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) + 4 \, e^{x} \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - e^{x} \log \left (x^{2}\right ) + 8 \, e^{x} \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, e^{x}\right )}}{x^{2} \log \left (x^{2}\right ) \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - x^{2} \log \left (x^{2}\right ) \log \left (x\right )^{2} \log \left (\log \left (x^{2}\right ) + 4\right ) + 2 \, x^{2} \log \left (x^{2}\right ) \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 2 \, x^{2} \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) + 4 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, x^{2} \log \left (x\right )^{2} \log \left (\log \left (x^{2}\right ) + 4\right ) - 2 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (x\right ) + 2 \, x^{2} \log \left (x\right )^{2} + 8 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 8 \, x^{2} \log \left (x\right ) \log \left (\log \left (x^{2}\right ) + 4\right ) - 4 \, x^{2} \log \left (x + \log \left (\log \left (x^{2}\right ) + 4\right )\right ) + 4 \, x^{2} \log \left (x\right )} \]
integrate(((((-6+3*x)*exp(x)*log(x^2)+(12*x-24)*exp(x))*log(4+log(x^2))+(3 *x^2-6*x)*exp(x)*log(x^2)+(12*x^2-24*x)*exp(x))*log(x/(log(4+log(x^2))+x)) +(-3*exp(x)*log(x^2)-12*exp(x))*log(4+log(x^2))+6*exp(x))/((x^3*log(x^2)+4 *x^3)*log(4+log(x^2))+x^4*log(x^2)+4*x^4)/log(x/(log(4+log(x^2))+x))^2,x, algorithm=\
-3*(e^x*log(x^2)*log(x)*log(log(x^2) + 4) + 2*e^x*log(x^2)*log(log(x^2) + 4) + 4*e^x*log(x)*log(log(x^2) + 4) - e^x*log(x^2) + 8*e^x*log(log(x^2) + 4) - 4*e^x)/(x^2*log(x^2)*log(x + log(log(x^2) + 4))*log(x)*log(log(x^2) + 4) - x^2*log(x^2)*log(x)^2*log(log(x^2) + 4) + 2*x^2*log(x^2)*log(x + log (log(x^2) + 4))*log(log(x^2) + 4) - 2*x^2*log(x^2)*log(x)*log(log(x^2) + 4 ) + 4*x^2*log(x + log(log(x^2) + 4))*log(x)*log(log(x^2) + 4) - 4*x^2*log( x)^2*log(log(x^2) + 4) - 2*x^2*log(x + log(log(x^2) + 4))*log(x) + 2*x^2*l og(x)^2 + 8*x^2*log(x + log(log(x^2) + 4))*log(log(x^2) + 4) - 8*x^2*log(x )*log(log(x^2) + 4) - 4*x^2*log(x + log(log(x^2) + 4)) + 4*x^2*log(x))
Timed out. \[ \int \frac {6 e^x+\left (-12 e^x-3 e^x \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )+\left (e^x \left (-24 x+12 x^2\right )+e^x \left (-6 x+3 x^2\right ) \log \left (x^2\right )+\left (e^x (-24+12 x)+e^x (-6+3 x) \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log \left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )}{\left (4 x^4+x^4 \log \left (x^2\right )+\left (4 x^3+x^3 \log \left (x^2\right )\right ) \log \left (4+\log \left (x^2\right )\right )\right ) \log ^2\left (\frac {x}{x+\log \left (4+\log \left (x^2\right )\right )}\right )} \, dx=\int -\frac {\ln \left (\ln \left (x^2\right )+4\right )\,\left (12\,{\mathrm {e}}^x+3\,\ln \left (x^2\right )\,{\mathrm {e}}^x\right )-6\,{\mathrm {e}}^x+\ln \left (\frac {x}{x+\ln \left (\ln \left (x^2\right )+4\right )}\right )\,\left ({\mathrm {e}}^x\,\left (24\,x-12\,x^2\right )-\ln \left (\ln \left (x^2\right )+4\right )\,\left ({\mathrm {e}}^x\,\left (12\,x-24\right )+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (3\,x-6\right )\right )+\ln \left (x^2\right )\,{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\right )}{{\ln \left (\frac {x}{x+\ln \left (\ln \left (x^2\right )+4\right )}\right )}^2\,\left (x^4\,\ln \left (x^2\right )+\ln \left (\ln \left (x^2\right )+4\right )\,\left (x^3\,\ln \left (x^2\right )+4\,x^3\right )+4\,x^4\right )} \,d x \]
int(-(log(log(x^2) + 4)*(12*exp(x) + 3*log(x^2)*exp(x)) - 6*exp(x) + log(x /(x + log(log(x^2) + 4)))*(exp(x)*(24*x - 12*x^2) - log(log(x^2) + 4)*(exp (x)*(12*x - 24) + log(x^2)*exp(x)*(3*x - 6)) + log(x^2)*exp(x)*(6*x - 3*x^ 2)))/(log(x/(x + log(log(x^2) + 4)))^2*(x^4*log(x^2) + log(log(x^2) + 4)*( x^3*log(x^2) + 4*x^3) + 4*x^4)),x)
int(-(log(log(x^2) + 4)*(12*exp(x) + 3*log(x^2)*exp(x)) - 6*exp(x) + log(x /(x + log(log(x^2) + 4)))*(exp(x)*(24*x - 12*x^2) - log(log(x^2) + 4)*(exp (x)*(12*x - 24) + log(x^2)*exp(x)*(3*x - 6)) + log(x^2)*exp(x)*(6*x - 3*x^ 2)))/(log(x/(x + log(log(x^2) + 4)))^2*(x^4*log(x^2) + log(log(x^2) + 4)*( x^3*log(x^2) + 4*x^3) + 4*x^4)), x)