Integrand size = 181, antiderivative size = 29 \[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\left (5+\log \left (\left (-e^{e^x}+2 x-\frac {16 e^{4 x}}{\log (x)}\right )^2\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(29)=58\).
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\log \left (\frac {\left (16 e^{4 x}+\left (e^{e^x}-2 x\right ) \log (x)\right )^2}{\log ^2(x)}\right ) \left (10+\log \left (\frac {\left (16 e^{4 x}+\left (e^{e^x}-2 x\right ) \log (x)\right )^2}{\log ^2(x)}\right )\right ) \]
Integrate[(-320*E^(4*x) + 1280*E^(4*x)*x*Log[x] - 40*x*Log[x]^2 + 20*E^(E^ x + x)*x*Log[x]^2 + (-64*E^(4*x) + 256*E^(4*x)*x*Log[x] - 8*x*Log[x]^2 + 4 *E^(E^x + x)*x*Log[x]^2)*Log[(256*E^(8*x) - 64*E^(4*x)*x*Log[x] + E^(2*E^x )*Log[x]^2 + 4*x^2*Log[x]^2 + E^E^x*(32*E^(4*x)*Log[x] - 4*x*Log[x]^2))/Lo g[x]^2])/(16*E^(4*x)*x*Log[x] + E^E^x*x*Log[x]^2 - 2*x^2*Log[x]^2),x]
Log[(16*E^(4*x) + (E^E^x - 2*x)*Log[x])^2/Log[x]^2]*(10 + Log[(16*E^(4*x) + (E^E^x - 2*x)*Log[x])^2/Log[x]^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.
| \(\displaystyle \int \frac {\left (-64 e^{4 x}+4 e^{x+e^x} x \log ^2(x)-8 x \log ^2(x)+256 e^{4 x} x \log (x)\right ) \log \left (\frac {4 x^2 \log ^2(x)+256 e^{8 x}+e^{2 e^x} \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )-64 e^{4 x} x \log (x)}{\log ^2(x)}\right )-320 e^{4 x}+20 e^{x+e^x} x \log ^2(x)-40 x \log ^2(x)+1280 e^{4 x} x \log (x)}{-2 x^2 \log ^2(x)+e^{e^x} x \log ^2(x)+16 e^{4 x} x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 \left (16 e^{4 x}-\left (\left (e^{x+e^x}-2\right ) x \log ^2(x)\right )-64 e^{4 x} x \log (x)\right ) \left (-\log \left (\frac {\left (16 e^{4 x}+\left (e^{e^x}-2 x\right ) \log (x)\right )^2}{\log ^2(x)}\right )-5\right )}{x \log (x) \left (16 e^{4 x}+\left (e^{e^x}-2 x\right ) \log (x)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {\left (\left (2-e^{x+e^x}\right ) x \log ^2(x)-64 e^{4 x} x \log (x)+16 e^{4 x}\right ) \left (\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )+5\right )}{x \log (x) \left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {\left (\left (2-e^{x+e^x}\right ) x \log ^2(x)-64 e^{4 x} x \log (x)+16 e^{4 x}\right ) \left (\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )+5\right )}{x \log (x) \left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (\frac {\left (8 \log (x) x^2-4 e^{e^x} \log (x) x+e^{x+e^x} \log (x) x-2 \log (x) x-2 x+e^{e^x}\right ) \left (\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )+5\right )}{x \left (-e^{e^x} \log (x)+2 x \log (x)-16 e^{4 x}\right )}-\frac {(4 x \log (x)-1) \left (\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )+5\right )}{x \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (16 x^2-4 \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right ) x-20 x+5 \log (\log (x))-8 \operatorname {LogIntegral}(x)+10 \int \frac {1}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+8 \int \frac {e^{e^x}}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx-5 \int \frac {e^{e^x}}{x \left (e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}\right )}dx+10 \int \frac {\log (x)}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+20 \int \frac {e^{e^x} \log (x)}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx-5 \int \frac {e^{x+e^x} \log (x)}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx-32 \int \frac {e^{e^x} x \log (x)}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+8 \int \frac {e^{x+e^x} x \log (x)}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+16 \int \frac {x}{-e^{e^x} \log (x)+2 x \log (x)-16 e^{4 x}}dx+56 \int \frac {x \log (x)}{-e^{e^x} \log (x)+2 x \log (x)-16 e^{4 x}}dx-64 \int \frac {x^2 \log (x)}{-e^{e^x} \log (x)+2 x \log (x)-16 e^{4 x}}dx+\int \frac {\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{x \log (x)}dx+2 \int \frac {\log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx-\int \frac {e^{e^x} \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{x \left (e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}\right )}dx+2 \int \frac {\log (x) \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+4 \int \frac {e^{e^x} \log (x) \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx-\int \frac {e^{x+e^x} \log (x) \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{e^{e^x} \log (x)-2 x \log (x)+16 e^{4 x}}dx+8 \int \frac {x \log (x) \log \left (\frac {\left (\left (e^{e^x}-2 x\right ) \log (x)+16 e^{4 x}\right )^2}{\log ^2(x)}\right )}{-e^{e^x} \log (x)+2 x \log (x)-16 e^{4 x}}dx\right )\) |
Int[(-320*E^(4*x) + 1280*E^(4*x)*x*Log[x] - 40*x*Log[x]^2 + 20*E^(E^x + x) *x*Log[x]^2 + (-64*E^(4*x) + 256*E^(4*x)*x*Log[x] - 8*x*Log[x]^2 + 4*E^(E^ x + x)*x*Log[x]^2)*Log[(256*E^(8*x) - 64*E^(4*x)*x*Log[x] + E^(2*E^x)*Log[ x]^2 + 4*x^2*Log[x]^2 + E^E^x*(32*E^(4*x)*Log[x] - 4*x*Log[x]^2))/Log[x]^2 ])/(16*E^(4*x)*x*Log[x] + E^E^x*x*Log[x]^2 - 2*x^2*Log[x]^2),x]
3.1.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
\[\int \frac {\left (4 x \,{\mathrm e}^{x} \ln \left (x \right )^{2} {\mathrm e}^{{\mathrm e}^{x}}-8 x \ln \left (x \right )^{2}+256 x \,{\mathrm e}^{4 x} \ln \left (x \right )-64 \,{\mathrm e}^{4 x}\right ) \ln \left (\frac {\ln \left (x \right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (-4 x \ln \left (x \right )^{2}+32 \,{\mathrm e}^{4 x} \ln \left (x \right )\right ) {\mathrm e}^{{\mathrm e}^{x}}+4 x^{2} \ln \left (x \right )^{2}-64 x \,{\mathrm e}^{4 x} \ln \left (x \right )+256 \,{\mathrm e}^{8 x}}{\ln \left (x \right )^{2}}\right )+20 x \,{\mathrm e}^{x} \ln \left (x \right )^{2} {\mathrm e}^{{\mathrm e}^{x}}-40 x \ln \left (x \right )^{2}+1280 x \,{\mathrm e}^{4 x} \ln \left (x \right )-320 \,{\mathrm e}^{4 x}}{x \ln \left (x \right )^{2} {\mathrm e}^{{\mathrm e}^{x}}-2 x^{2} \ln \left (x \right )^{2}+16 x \,{\mathrm e}^{4 x} \ln \left (x \right )}d x\]
int(((4*x*exp(x)*ln(x)^2*exp(exp(x))-8*x*ln(x)^2+256*x*exp(x)^4*ln(x)-64*e xp(x)^4)*ln((ln(x)^2*exp(exp(x))^2+(-4*x*ln(x)^2+32*exp(x)^4*ln(x))*exp(ex p(x))+4*x^2*ln(x)^2-64*x*exp(x)^4*ln(x)+256*exp(x)^8)/ln(x)^2)+20*x*exp(x) *ln(x)^2*exp(exp(x))-40*x*ln(x)^2+1280*x*exp(x)^4*ln(x)-320*exp(x)^4)/(x*l n(x)^2*exp(exp(x))-2*x^2*ln(x)^2+16*x*exp(x)^4*ln(x)),x)
int(((4*x*exp(x)*ln(x)^2*exp(exp(x))-8*x*ln(x)^2+256*x*exp(x)^4*ln(x)-64*e xp(x)^4)*ln((ln(x)^2*exp(exp(x))^2+(-4*x*ln(x)^2+32*exp(x)^4*ln(x))*exp(ex p(x))+4*x^2*ln(x)^2-64*x*exp(x)^4*ln(x)+256*exp(x)^8)/ln(x)^2)+20*x*exp(x) *ln(x)^2*exp(exp(x))-40*x*ln(x)^2+1280*x*exp(x)^4*ln(x)-320*exp(x)^4)/(x*l n(x)^2*exp(exp(x))-2*x^2*ln(x)^2+16*x*exp(x)^4*ln(x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 5.48 \[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\log \left (\frac {{\left (4 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right )^{2} - 64 \, x e^{\left (6 \, x\right )} \log \left (x\right ) + e^{\left (2 \, x + 2 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, {\left (x e^{x} \log \left (x\right )^{2} - 8 \, e^{\left (5 \, x\right )} \log \left (x\right )\right )} e^{\left (x + e^{x}\right )} + 256 \, e^{\left (10 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{\log \left (x\right )^{2}}\right )^{2} + 10 \, \log \left (\frac {{\left (4 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right )^{2} - 64 \, x e^{\left (6 \, x\right )} \log \left (x\right ) + e^{\left (2 \, x + 2 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, {\left (x e^{x} \log \left (x\right )^{2} - 8 \, e^{\left (5 \, x\right )} \log \left (x\right )\right )} e^{\left (x + e^{x}\right )} + 256 \, e^{\left (10 \, x\right )}\right )} e^{\left (-2 \, x\right )}}{\log \left (x\right )^{2}}\right ) \]
integrate(((4*x*exp(x)*log(x)^2*exp(exp(x))-8*x*log(x)^2+256*x*exp(x)^4*lo g(x)-64*exp(x)^4)*log((log(x)^2*exp(exp(x))^2+(-4*x*log(x)^2+32*exp(x)^4*l og(x))*exp(exp(x))+4*x^2*log(x)^2-64*x*exp(x)^4*log(x)+256*exp(x)^8)/log(x )^2)+20*x*exp(x)*log(x)^2*exp(exp(x))-40*x*log(x)^2+1280*x*exp(x)^4*log(x) -320*exp(x)^4)/(x*log(x)^2*exp(exp(x))-2*x^2*log(x)^2+16*x*exp(x)^4*log(x) ),x, algorithm=\
log((4*x^2*e^(2*x)*log(x)^2 - 64*x*e^(6*x)*log(x) + e^(2*x + 2*e^x)*log(x) ^2 - 4*(x*e^x*log(x)^2 - 8*e^(5*x)*log(x))*e^(x + e^x) + 256*e^(10*x))*e^( -2*x)/log(x)^2)^2 + 10*log((4*x^2*e^(2*x)*log(x)^2 - 64*x*e^(6*x)*log(x) + e^(2*x + 2*e^x)*log(x)^2 - 4*(x*e^x*log(x)^2 - 8*e^(5*x)*log(x))*e^(x + e ^x) + 256*e^(10*x))*e^(-2*x)/log(x)^2)
Timed out. \[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\text {Timed out} \]
integrate(((4*x*exp(x)*ln(x)**2*exp(exp(x))-8*x*ln(x)**2+256*x*exp(x)**4*l n(x)-64*exp(x)**4)*ln((ln(x)**2*exp(exp(x))**2+(-4*x*ln(x)**2+32*exp(x)**4 *ln(x))*exp(exp(x))+4*x**2*ln(x)**2-64*x*exp(x)**4*ln(x)+256*exp(x)**8)/ln (x)**2)+20*x*exp(x)*ln(x)**2*exp(exp(x))-40*x*ln(x)**2+1280*x*exp(x)**4*ln (x)-320*exp(x)**4)/(x*ln(x)**2*exp(exp(x))-2*x**2*ln(x)**2+16*x*exp(x)**4* ln(x)),x)
\[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\int { -\frac {4 \, {\left (5 \, x e^{\left (x + e^{x}\right )} \log \left (x\right )^{2} + 320 \, x e^{\left (4 \, x\right )} \log \left (x\right ) - 10 \, x \log \left (x\right )^{2} + {\left (x e^{\left (x + e^{x}\right )} \log \left (x\right )^{2} + 64 \, x e^{\left (4 \, x\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} - 16 \, e^{\left (4 \, x\right )}\right )} \log \left (\frac {4 \, x^{2} \log \left (x\right )^{2} - 64 \, x e^{\left (4 \, x\right )} \log \left (x\right ) + e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, {\left (x \log \left (x\right )^{2} - 8 \, e^{\left (4 \, x\right )} \log \left (x\right )\right )} e^{\left (e^{x}\right )} + 256 \, e^{\left (8 \, x\right )}}{\log \left (x\right )^{2}}\right ) - 80 \, e^{\left (4 \, x\right )}\right )}}{2 \, x^{2} \log \left (x\right )^{2} - x e^{\left (e^{x}\right )} \log \left (x\right )^{2} - 16 \, x e^{\left (4 \, x\right )} \log \left (x\right )} \,d x } \]
integrate(((4*x*exp(x)*log(x)^2*exp(exp(x))-8*x*log(x)^2+256*x*exp(x)^4*lo g(x)-64*exp(x)^4)*log((log(x)^2*exp(exp(x))^2+(-4*x*log(x)^2+32*exp(x)^4*l og(x))*exp(exp(x))+4*x^2*log(x)^2-64*x*exp(x)^4*log(x)+256*exp(x)^8)/log(x )^2)+20*x*exp(x)*log(x)^2*exp(exp(x))-40*x*log(x)^2+1280*x*exp(x)^4*log(x) -320*exp(x)^4)/(x*log(x)^2*exp(exp(x))-2*x^2*log(x)^2+16*x*exp(x)^4*log(x) ),x, algorithm=\
-4*integrate((5*x*e^(x + e^x)*log(x)^2 + 320*x*e^(4*x)*log(x) - 10*x*log(x )^2 + (x*e^(x + e^x)*log(x)^2 + 64*x*e^(4*x)*log(x) - 2*x*log(x)^2 - 16*e^ (4*x))*log((4*x^2*log(x)^2 - 64*x*e^(4*x)*log(x) + e^(2*e^x)*log(x)^2 - 4* (x*log(x)^2 - 8*e^(4*x)*log(x))*e^(e^x) + 256*e^(8*x))/log(x)^2) - 80*e^(4 *x))/(2*x^2*log(x)^2 - x*e^(e^x)*log(x)^2 - 16*x*e^(4*x)*log(x)), x)
\[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx=\int { -\frac {4 \, {\left (5 \, x e^{\left (x + e^{x}\right )} \log \left (x\right )^{2} + 320 \, x e^{\left (4 \, x\right )} \log \left (x\right ) - 10 \, x \log \left (x\right )^{2} + {\left (x e^{\left (x + e^{x}\right )} \log \left (x\right )^{2} + 64 \, x e^{\left (4 \, x\right )} \log \left (x\right ) - 2 \, x \log \left (x\right )^{2} - 16 \, e^{\left (4 \, x\right )}\right )} \log \left (\frac {4 \, x^{2} \log \left (x\right )^{2} - 64 \, x e^{\left (4 \, x\right )} \log \left (x\right ) + e^{\left (2 \, e^{x}\right )} \log \left (x\right )^{2} - 4 \, {\left (x \log \left (x\right )^{2} - 8 \, e^{\left (4 \, x\right )} \log \left (x\right )\right )} e^{\left (e^{x}\right )} + 256 \, e^{\left (8 \, x\right )}}{\log \left (x\right )^{2}}\right ) - 80 \, e^{\left (4 \, x\right )}\right )}}{2 \, x^{2} \log \left (x\right )^{2} - x e^{\left (e^{x}\right )} \log \left (x\right )^{2} - 16 \, x e^{\left (4 \, x\right )} \log \left (x\right )} \,d x } \]
integrate(((4*x*exp(x)*log(x)^2*exp(exp(x))-8*x*log(x)^2+256*x*exp(x)^4*lo g(x)-64*exp(x)^4)*log((log(x)^2*exp(exp(x))^2+(-4*x*log(x)^2+32*exp(x)^4*l og(x))*exp(exp(x))+4*x^2*log(x)^2-64*x*exp(x)^4*log(x)+256*exp(x)^8)/log(x )^2)+20*x*exp(x)*log(x)^2*exp(exp(x))-40*x*log(x)^2+1280*x*exp(x)^4*log(x) -320*exp(x)^4)/(x*log(x)^2*exp(exp(x))-2*x^2*log(x)^2+16*x*exp(x)^4*log(x) ),x, algorithm=\
integrate(-4*(5*x*e^(x + e^x)*log(x)^2 + 320*x*e^(4*x)*log(x) - 10*x*log(x )^2 + (x*e^(x + e^x)*log(x)^2 + 64*x*e^(4*x)*log(x) - 2*x*log(x)^2 - 16*e^ (4*x))*log((4*x^2*log(x)^2 - 64*x*e^(4*x)*log(x) + e^(2*e^x)*log(x)^2 - 4* (x*log(x)^2 - 8*e^(4*x)*log(x))*e^(e^x) + 256*e^(8*x))/log(x)^2) - 80*e^(4 *x))/(2*x^2*log(x)^2 - x*e^(e^x)*log(x)^2 - 16*x*e^(4*x)*log(x)), x)
Time = 11.81 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \frac {-320 e^{4 x}+1280 e^{4 x} x \log (x)-40 x \log ^2(x)+20 e^{e^x+x} x \log ^2(x)+\left (-64 e^{4 x}+256 e^{4 x} x \log (x)-8 x \log ^2(x)+4 e^{e^x+x} x \log ^2(x)\right ) \log \left (\frac {256 e^{8 x}-64 e^{4 x} x \log (x)+e^{2 e^x} \log ^2(x)+4 x^2 \log ^2(x)+e^{e^x} \left (32 e^{4 x} \log (x)-4 x \log ^2(x)\right )}{\log ^2(x)}\right )}{16 e^{4 x} x \log (x)+e^{e^x} x \log ^2(x)-2 x^2 \log ^2(x)} \, dx={\ln \left (\frac {256\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,{\ln \left (x\right )}^2+4\,x^2\,{\ln \left (x\right )}^2-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (4\,x\,{\ln \left (x\right )}^2-32\,{\mathrm {e}}^{4\,x}\,\ln \left (x\right )\right )-64\,x\,{\mathrm {e}}^{4\,x}\,\ln \left (x\right )}{{\ln \left (x\right )}^2}\right )}^2+20\,\ln \left (\frac {16\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{{\mathrm {e}}^x}\,\ln \left (x\right )-2\,x\,\ln \left (x\right )}{\ln \left (x\right )}\right ) \]
int(-(320*exp(4*x) + log((256*exp(8*x) + exp(2*exp(x))*log(x)^2 + 4*x^2*lo g(x)^2 - exp(exp(x))*(4*x*log(x)^2 - 32*exp(4*x)*log(x)) - 64*x*exp(4*x)*l og(x))/log(x)^2)*(64*exp(4*x) + 8*x*log(x)^2 - 256*x*exp(4*x)*log(x) - 4*x *exp(exp(x))*exp(x)*log(x)^2) + 40*x*log(x)^2 - 1280*x*exp(4*x)*log(x) - 2 0*x*exp(exp(x))*exp(x)*log(x)^2)/(x*exp(exp(x))*log(x)^2 - 2*x^2*log(x)^2 + 16*x*exp(4*x)*log(x)),x)