Integrand size = 132, antiderivative size = 32 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\log \left (3+e^{\frac {e^{-x} \left (3+\frac {x+x^2}{\log \left (4 x^2\right )}\right )}{\log (\log (4))}}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(32)=64\).
Time = 0.46 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\frac {3 e^{-x}}{\log (\log (4))}+\frac {\frac {e^{-x} x (1+x)}{\log \left (4 x^2\right )}+\log \left (1+3 e^{-\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}}\right ) \log (\log (4))}{\log (\log (4))} \]
Integrate[(E^((x + x^2 + 3*Log[4*x^2])/(E^x*Log[4*x^2]*Log[Log[4]]))*(-2 - 2*x + (1 + x - x^2)*Log[4*x^2] - 3*Log[4*x^2]^2))/(3*E^x*Log[4*x^2]^2*Log [Log[4]] + E^(x + (x + x^2 + 3*Log[4*x^2])/(E^x*Log[4*x^2]*Log[Log[4]]))*L og[4*x^2]^2*Log[Log[4]]),x]
3/(E^x*Log[Log[4]]) + ((x*(1 + x))/(E^x*Log[4*x^2]) + Log[1 + 3/E^((x + x^ 2 + 3*Log[4*x^2])/(E^x*Log[4*x^2]*Log[Log[4]]))]*Log[Log[4]])/Log[Log[4]]
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.
| \(\displaystyle \int \frac {\left (-3 \log ^2\left (4 x^2\right )+\left (-x^2+x+1\right ) \log \left (4 x^2\right )-2 x-2\right ) \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}\right )}{\log (\log (4)) \log ^2\left (4 x^2\right ) \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}+x\right )+3 e^x \log (\log (4)) \log ^2\left (4 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \log ^2\left (4 x^2\right ) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}-\frac {2 \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \log ^2\left (4 x^2\right ) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}-\frac {x^2 \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \log \left (4 x^2\right ) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}+\frac {x \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \log \left (4 x^2\right ) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}-\frac {3 \exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}+\frac {\exp \left (\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}-x\right )}{\log (\log (4)) \log \left (4 x^2\right ) \left (\exp \left (\frac {e^{-x} x^2}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {e^{-x} x}{\log (\log (4)) \log \left (4 x^2\right )}+\frac {3 e^{-x}}{\log (\log (4))}\right )+3\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-x} \left (-3 \log ^2\left (4 x^2\right )+\left (-x^2+x+1\right ) \log \left (4 x^2\right )-2 (x+1)\right )}{\log (\log (4)) \log ^2\left (4 x^2\right ) \left (3 \exp \left (-\frac {e^{-x} \left (x^2+3 \log \left (4 x^2\right )+x\right )}{\log (\log (4)) \log \left (4 x^2\right )}\right )+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {e^{-x} \left (3 \log ^2\left (4 x^2\right )-\left (-x^2+x+1\right ) \log \left (4 x^2\right )+2 (x+1)\right )}{\left (3\ 4^{-\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}} e^{-\frac {e^{-x} \left (x^2+x\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (x^2\right )^{-\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}}+1\right ) \log ^2\left (4 x^2\right )}dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {e^{-x} \left (3 \log ^2\left (4 x^2\right )-\left (-x^2+x+1\right ) \log \left (4 x^2\right )+2 (x+1)\right )}{\left (3\ 4^{-\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}} e^{-\frac {e^{-x} \left (x^2+x\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (x^2\right )^{-\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}}+1\right ) \log ^2\left (4 x^2\right )}dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {e^{-x} \left (\log \left (4 x^2\right ) x^2-\log \left (4 x^2\right ) x+2 x+3 \log ^2\left (4 x^2\right )-\log \left (4 x^2\right )+2\right )}{\log ^2\left (4 x^2\right )}-\frac {3 e^{-x} \left (\log \left (4 x^2\right ) x^2-\log \left (4 x^2\right ) x+2 x+3 \log ^2\left (4 x^2\right )-\log \left (4 x^2\right )+2\right )}{\left (64^{\frac {e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}} e^{\frac {e^{-x} x (x+1)}{\log \left (4 x^2\right ) \log (\log (4))}} \left (x^2\right )^{\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}}+3\right ) \log ^2\left (4 x^2\right )}\right )dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {\int \left (\frac {e^{-x} \left (\log \left (4 x^2\right ) x^2-\log \left (4 x^2\right ) x+2 x+3 \log ^2\left (4 x^2\right )-\log \left (4 x^2\right )+2\right )}{\log ^2\left (4 x^2\right )}-\frac {3 e^{-x} \left (\log \left (4 x^2\right ) x^2-\log \left (4 x^2\right ) x+2 x+3 \log ^2\left (4 x^2\right )-\log \left (4 x^2\right )+2\right )}{\left (64^{\frac {e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}} e^{\frac {e^{-x} x (x+1)}{\log \left (4 x^2\right ) \log (\log (4))}} \left (x^2\right )^{\frac {3 e^{-x}}{\log \left (4 x^2\right ) \log (\log (4))}}+3\right ) \log ^2\left (4 x^2\right )}\right )dx}{\log (\log (4))}\) |
Int[(E^((x + x^2 + 3*Log[4*x^2])/(E^x*Log[4*x^2]*Log[Log[4]]))*(-2 - 2*x + (1 + x - x^2)*Log[4*x^2] - 3*Log[4*x^2]^2))/(3*E^x*Log[4*x^2]^2*Log[Log[4 ]] + E^(x + (x + x^2 + 3*Log[4*x^2])/(E^x*Log[4*x^2]*Log[Log[4]]))*Log[4*x ^2]^2*Log[Log[4]]),x]
3.13.86.3.1 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]]
Time = 41.54 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19
| method | result | size |
| parallelrisch | \(\ln \left ({\mathrm e}^{\frac {\left (3 \ln \left (4 x^{2}\right )+x^{2}+x \right ) {\mathrm e}^{-x}}{\ln \left (4 x^{2}\right ) \ln \left (2 \ln \left (2\right )\right )}}+3\right )\) | \(38\) |
| risch | \(\frac {3 \,{\mathrm e}^{-x}}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}+\frac {2 i \left (1+x \right ) x \,{\mathrm e}^{-x}}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (2\right )+4 i \ln \left (x \right )\right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )}-\frac {\left (6 \ln \left (2\right )+6 \ln \left (x \right )-\frac {3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}+x^{2}+x \right ) {\mathrm e}^{-x}}{\left (2 \ln \left (2\right )+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )}+\ln \left ({\mathrm e}^{\frac {\left (-3 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 x^{2}+12 \ln \left (x \right )+12 \ln \left (2\right )+2 x \right ) {\mathrm e}^{-x}}{\left (4 \ln \left (2\right )+4 \ln \left (x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}\right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right )}}+3\right )\) | \(329\) |
int((-3*ln(4*x^2)^2+(-x^2+x+1)*ln(4*x^2)-2*x-2)*exp((3*ln(4*x^2)+x^2+x)/ex p(x)/ln(4*x^2)/ln(2*ln(2)))/(exp(x)*ln(4*x^2)^2*ln(2*ln(2))*exp((3*ln(4*x^ 2)+x^2+x)/exp(x)/ln(4*x^2)/ln(2*ln(2)))+3*exp(x)*ln(4*x^2)^2*ln(2*ln(2))), x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=-x + \log \left (3 \, e^{x} + e^{\left (\frac {{\left (x e^{x} \log \left (4 \, x^{2}\right ) \log \left (2 \, \log \left (2\right )\right ) + x^{2} + x + 3 \, \log \left (4 \, x^{2}\right )\right )} e^{\left (-x\right )}}{\log \left (4 \, x^{2}\right ) \log \left (2 \, \log \left (2\right )\right )}\right )}\right ) \]
integrate((-3*log(4*x^2)^2+(-x^2+x+1)*log(4*x^2)-2*x-2)*exp((3*log(4*x^2)+ x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))/(exp(x)*log(4*x^2)^2*log(2*log(2)) *exp((3*log(4*x^2)+x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))+3*exp(x)*log(4* x^2)^2*log(2*log(2))),x, algorithm=\
-x + log(3*e^x + e^((x*e^x*log(4*x^2)*log(2*log(2)) + x^2 + x + 3*log(4*x^ 2))*e^(-x)/(log(4*x^2)*log(2*log(2)))))
Time = 0.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\log {\left (e^{\frac {\left (x^{2} + x + 3 \log {\left (4 x^{2} \right )}\right ) e^{- x}}{\log {\left (4 x^{2} \right )} \log {\left (2 \log {\left (2 \right )} \right )}}} + 3 \right )} \]
integrate((-3*ln(4*x**2)**2+(-x**2+x+1)*ln(4*x**2)-2*x-2)*exp((3*ln(4*x**2 )+x**2+x)/exp(x)/ln(4*x**2)/ln(2*ln(2)))/(exp(x)*ln(4*x**2)**2*ln(2*ln(2)) *exp((3*ln(4*x**2)+x**2+x)/exp(x)/ln(4*x**2)/ln(2*ln(2)))+3*exp(x)*ln(4*x* *2)**2*ln(2*ln(2))),x)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (32) = 64\).
Time = 0.50 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.31 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\frac {{\left (x + 6 \, \log \left (2\right ) + 6 \, \log \left (x\right )\right )} e^{\left (-x\right )}}{2 \, {\left (\log \left (2\right )^{2} + {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + \log \left (2\right ) \log \left (\log \left (2\right )\right )\right )}} + \log \left ({\left (e^{\left (\frac {x^{2}}{2 \, {\left ({\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} e^{x} \log \left (x\right ) + {\left (\log \left (2\right )^{2} + \log \left (2\right ) \log \left (\log \left (2\right )\right )\right )} e^{x}\right )}} + \frac {x}{2 \, {\left ({\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} e^{x} \log \left (x\right ) + {\left (\log \left (2\right )^{2} + \log \left (2\right ) \log \left (\log \left (2\right )\right )\right )} e^{x}\right )}} + \frac {3 \, e^{\left (-x\right )}}{\log \left (2\right ) + \log \left (\log \left (2\right )\right )}\right )} + 3\right )} e^{\left (-\frac {x}{2 \, {\left ({\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} e^{x} \log \left (x\right ) + {\left (\log \left (2\right )^{2} + \log \left (2\right ) \log \left (\log \left (2\right )\right )\right )} e^{x}\right )}} - \frac {3 \, e^{\left (-x\right )}}{\log \left (2\right ) + \log \left (\log \left (2\right )\right )}\right )}\right ) \]
integrate((-3*log(4*x^2)^2+(-x^2+x+1)*log(4*x^2)-2*x-2)*exp((3*log(4*x^2)+ x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))/(exp(x)*log(4*x^2)^2*log(2*log(2)) *exp((3*log(4*x^2)+x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))+3*exp(x)*log(4* x^2)^2*log(2*log(2))),x, algorithm=\
1/2*(x + 6*log(2) + 6*log(x))*e^(-x)/(log(2)^2 + (log(2) + log(log(2)))*lo g(x) + log(2)*log(log(2))) + log((e^(1/2*x^2/((log(2) + log(log(2)))*e^x*l og(x) + (log(2)^2 + log(2)*log(log(2)))*e^x) + 1/2*x/((log(2) + log(log(2) ))*e^x*log(x) + (log(2)^2 + log(2)*log(log(2)))*e^x) + 3*e^(-x)/(log(2) + log(log(2)))) + 3)*e^(-1/2*x/((log(2) + log(log(2)))*e^x*log(x) + (log(2)^ 2 + log(2)*log(log(2)))*e^x) - 3*e^(-x)/(log(2) + log(log(2)))))
\[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\int { -\frac {{\left ({\left (x^{2} - x - 1\right )} \log \left (4 \, x^{2}\right ) + 3 \, \log \left (4 \, x^{2}\right )^{2} + 2 \, x + 2\right )} e^{\left (\frac {{\left (x^{2} + x + 3 \, \log \left (4 \, x^{2}\right )\right )} e^{\left (-x\right )}}{\log \left (4 \, x^{2}\right ) \log \left (2 \, \log \left (2\right )\right )}\right )}}{e^{\left (x + \frac {{\left (x^{2} + x + 3 \, \log \left (4 \, x^{2}\right )\right )} e^{\left (-x\right )}}{\log \left (4 \, x^{2}\right ) \log \left (2 \, \log \left (2\right )\right )}\right )} \log \left (4 \, x^{2}\right )^{2} \log \left (2 \, \log \left (2\right )\right ) + 3 \, e^{x} \log \left (4 \, x^{2}\right )^{2} \log \left (2 \, \log \left (2\right )\right )} \,d x } \]
integrate((-3*log(4*x^2)^2+(-x^2+x+1)*log(4*x^2)-2*x-2)*exp((3*log(4*x^2)+ x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))/(exp(x)*log(4*x^2)^2*log(2*log(2)) *exp((3*log(4*x^2)+x^2+x)/exp(x)/log(4*x^2)/log(2*log(2)))+3*exp(x)*log(4* x^2)^2*log(2*log(2))),x, algorithm=\
Time = 13.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.34 \[ \int \frac {e^{\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \left (-2-2 x+\left (1+x-x^2\right ) \log \left (4 x^2\right )-3 \log ^2\left (4 x^2\right )\right )}{3 e^x \log ^2\left (4 x^2\right ) \log (\log (4))+e^{x+\frac {e^{-x} \left (x+x^2+3 \log \left (4 x^2\right )\right )}{\log \left (4 x^2\right ) \log (\log (4))}} \log ^2\left (4 x^2\right ) \log (\log (4))} \, dx=\ln \left ({\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,\ln \left (64\,x^6\right )}{\ln \left (4\right )\,\ln \left (\ln \left (2\right )\right )+2\,{\ln \left (2\right )}^2+\ln \left (2\,\ln \left (2\right )\right )\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{-x}}{\ln \left (4\right )\,\ln \left (\ln \left (2\right )\right )+2\,{\ln \left (2\right )}^2+\ln \left (2\,\ln \left (2\right )\right )\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{-x}}{\ln \left (4\right )\,\ln \left (\ln \left (2\right )\right )+2\,{\ln \left (2\right )}^2+\ln \left (2\,\ln \left (2\right )\right )\,\ln \left (x^2\right )}}+3\right ) \]
int(-(exp((exp(-x)*(x + 3*log(4*x^2) + x^2))/(log(2*log(2))*log(4*x^2)))*( 2*x + 3*log(4*x^2)^2 - log(4*x^2)*(x - x^2 + 1) + 2))/(3*log(2*log(2))*exp (x)*log(4*x^2)^2 + log(2*log(2))*exp((exp(-x)*(x + 3*log(4*x^2) + x^2))/(l og(2*log(2))*log(4*x^2)))*exp(x)*log(4*x^2)^2),x)