Integrand size = 291, antiderivative size = 31 \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=\frac {e^x}{x}-\log ^{\frac {x}{3}}\left (4+x-\frac {5 x}{x-\log (2)}\right ) \]
Timed out. \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=\text {\$Aborted} \]
Integrate[(E^x*(3*x^2 - 6*x^3 + 3*x^4 + (9*x - 3*x^2 - 6*x^3)*Log[2] + (-1 2 + 9*x + 3*x^2)*Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])] + Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]^(x/3)*(-x^5 + (-5*x^3 + 2*x ^4)*Log[2] - x^3*Log[2]^2 + (x^4 - x^5 + (3*x^3 + 2*x^4)*Log[2] + (-4*x^2 - x^3)*Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]*Log[Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]]))/((-3*x^4 + 3*x^5 + (-9*x^3 - 6*x ^4)*Log[2] + (12*x^2 + 3*x^3)*Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[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.
| \(\displaystyle \int \frac {e^x \left (3 x^4-6 x^3+3 x^2+\left (3 x^2+9 x-12\right ) \log ^2(2)+\left (-6 x^3-3 x^2+9 x\right ) \log (2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )+\left (-x^5-x^3 \log ^2(2)+\left (2 x^4-5 x^3\right ) \log (2)+\left (-x^5+x^4+\left (2 x^4+3 x^3\right ) \log (2)+\left (-x^3-4 x^2\right ) \log ^2(2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )\right ) \log ^{\frac {x}{3}}\left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )}{\left (3 x^5-3 x^4+\left (-6 x^4-9 x^3\right ) \log (2)+\left (3 x^3+12 x^2\right ) \log ^2(2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^x \left (3 x^4-6 x^3+3 x^2+\left (3 x^2+9 x-12\right ) \log ^2(2)+\left (-6 x^3-3 x^2+9 x\right ) \log (2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )+\left (-x^5-x^3 \log ^2(2)+\left (2 x^4-5 x^3\right ) \log (2)+\left (-x^5+x^4+\left (2 x^4+3 x^3\right ) \log (2)+\left (-x^3-4 x^2\right ) \log ^2(2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )\right ) \log ^{\frac {x}{3}}\left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )}{x^2 \left (3 x^3-3 x^2 (1+\log (4))-3 x (3-\log (2)) \log (2)+12 \log ^2(2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (3 x^4-6 x^3+3 x^2+\left (3 x^2+9 x-12\right ) \log ^2(2)+\left (-6 x^3-3 x^2+9 x\right ) \log (2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )+\left (-x^5-x^3 \log ^2(2)+\left (2 x^4-5 x^3\right ) \log (2)+\left (-x^5+x^4+\left (2 x^4+3 x^3\right ) \log (2)+\left (-x^3-4 x^2\right ) \log ^2(2)\right ) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )\right ) \log ^{\frac {x}{3}}\left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )}{x^2 \left (3 x^3-3 x^2 (1+\log (4))-3 x (3-\log (2)) \log (2)+12 \log ^2(2)\right ) \log \left (\frac {-x^2+x (1+\log (2))+\log (16)}{\log (2)-x}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^x (1-x) (x-\log (2)) \left (-x^2+x (1+\log (2))+\log (16)\right )}{x^2 \left (x^3-x^2 (1+\log (4))+x \left (\log ^2(2)-\log (8)\right )+4 \log ^2(2)\right )}+\frac {\left (-x^3-4 \log ^2(2) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )+x^2 (1+\log (4)) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )+x^2 \log (4)+3 x \left (1-\frac {\log (2)}{3}\right ) \log (2) \log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )+x^3 \left (-\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right ) \log \left (\log \left (\frac {-x^2+x+(x+4) \log (2)}{\log (2)-x}\right )\right )-x \log ^2(2) \left (1+\frac {\log (32)}{\log ^2(2)}\right )\right ) \log ^{\frac {x}{3}-1}\left (\frac {-x^2+x (1+\log (2))+\log (16)}{\log (2)-x}\right )}{3 \left (x^3-x^2 (1+\log (4))-x (3-\log (2)) \log (2)+4 \log ^2(2)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{3} \int \log ^{\frac {x}{3}-1}\left (\frac {-x^2+(1+\log (2)) x+\log (16)}{\log (2)-x}\right )dx-\frac {1}{3} \int \log ^{\frac {x}{3}}\left (-\frac {-x^2+(1+\log (2)) x+\log (16)}{x-\log (2)}\right ) \log \left (\log \left (-\frac {-x^2+(1+\log (2)) x+\log (16)}{x-\log (2)}\right )\right )dx+\frac {4}{3} \log ^2(2) \int \frac {\log ^{\frac {x}{3}-1}\left (\frac {-x^2+(1+\log (2)) x+\log (16)}{\log (2)-x}\right )}{x^3-(1+\log (4)) x^2-(3-\log (2)) \log (2) x+4 \log ^2(2)}dx+\frac {8}{3} \log (2) \int \frac {x \log ^{\frac {x}{3}-1}\left (\frac {-x^2+(1+\log (2)) x+\log (16)}{\log (2)-x}\right )}{-x^3+(1+\log (4)) x^2+(3-\log (2)) \log (2) x-4 \log ^2(2)}dx+\frac {1}{3} \int \frac {x^2 \log ^{\frac {x}{3}-1}\left (\frac {-x^2+(1+\log (2)) x+\log (16)}{\log (2)-x}\right )}{-x^3+(1+\log (4)) x^2+(3-\log (2)) \log (2) x-4 \log ^2(2)}dx+\frac {e^x}{x}\) |
Int[(E^x*(3*x^2 - 6*x^3 + 3*x^4 + (9*x - 3*x^2 - 6*x^3)*Log[2] + (-12 + 9* x + 3*x^2)*Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])] + Log[( x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]^(x/3)*(-x^5 + (-5*x^3 + 2*x^4)*Lo g[2] - x^3*Log[2]^2 + (x^4 - x^5 + (3*x^3 + 2*x^4)*Log[2] + (-4*x^2 - x^3) *Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]*Log[Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log[2])]]))/((-3*x^4 + 3*x^5 + (-9*x^3 - 6*x^4)*Lo g[2] + (12*x^2 + 3*x^3)*Log[2]^2)*Log[(x - x^2 + (4 + x)*Log[2])/(-x + Log [2])]),x]
3.23.94.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 4.90
\[\frac {{\mathrm e}^{x}}{x}-{\left (-\ln \left (\ln \left (2\right )-x \right )+\ln \left (\left (4+x \right ) \ln \left (2\right )-x^{2}+x \right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\left (4+x \right ) \ln \left (2\right )-x^{2}+x \right )}{\ln \left (2\right )-x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\left (4+x \right ) \ln \left (2\right )-x^{2}+x \right )}{\ln \left (2\right )-x}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (2\right )-x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\left (4+x \right ) \ln \left (2\right )-x^{2}+x \right )}{\ln \left (2\right )-x}\right )+\operatorname {csgn}\left (i \left (\left (4+x \right ) \ln \left (2\right )-x^{2}+x \right )\right )\right )}{2}\right )}^{\frac {x}{3}}\]
int(((((-x^3-4*x^2)*ln(2)^2+(2*x^4+3*x^3)*ln(2)-x^5+x^4)*ln(((4+x)*ln(2)-x ^2+x)/(ln(2)-x))*ln(ln(((4+x)*ln(2)-x^2+x)/(ln(2)-x)))-x^3*ln(2)^2+(2*x^4- 5*x^3)*ln(2)-x^5)*exp(1/3*x*ln(ln(((4+x)*ln(2)-x^2+x)/(ln(2)-x))))+((3*x^2 +9*x-12)*ln(2)^2+(-6*x^3-3*x^2+9*x)*ln(2)+3*x^4-6*x^3+3*x^2)*exp(x)*ln(((4 +x)*ln(2)-x^2+x)/(ln(2)-x)))/((3*x^3+12*x^2)*ln(2)^2+(-6*x^4-9*x^3)*ln(2)+ 3*x^5-3*x^4)/ln(((4+x)*ln(2)-x^2+x)/(ln(2)-x)),x)
exp(x)/x-(-ln(ln(2)-x)+ln((4+x)*ln(2)-x^2+x)-1/2*I*Pi*csgn(I/(ln(2)-x)*((4 +x)*ln(2)-x^2+x))*(-csgn(I/(ln(2)-x)*((4+x)*ln(2)-x^2+x))+csgn(I/(ln(2)-x) ))*(-csgn(I/(ln(2)-x)*((4+x)*ln(2)-x^2+x))+csgn(I*((4+x)*ln(2)-x^2+x))))^( 1/3*x)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=-\frac {x \log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right )^{\frac {1}{3} \, x} - e^{x}}{x} \]
integrate(((((-x^3-4*x^2)*log(2)^2+(2*x^4+3*x^3)*log(2)-x^5+x^4)*log(((4+x )*log(2)-x^2+x)/(log(2)-x))*log(log(((4+x)*log(2)-x^2+x)/(log(2)-x)))-x^3* log(2)^2+(2*x^4-5*x^3)*log(2)-x^5)*exp(1/3*x*log(log(((4+x)*log(2)-x^2+x)/ (log(2)-x))))+((3*x^2+9*x-12)*log(2)^2+(-6*x^3-3*x^2+9*x)*log(2)+3*x^4-6*x ^3+3*x^2)*exp(x)*log(((4+x)*log(2)-x^2+x)/(log(2)-x)))/((3*x^3+12*x^2)*log (2)^2+(-6*x^4-9*x^3)*log(2)+3*x^5-3*x^4)/log(((4+x)*log(2)-x^2+x)/(log(2)- x)),x, algorithm=\
Timed out. \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=\text {Timed out} \]
integrate(((((-x**3-4*x**2)*ln(2)**2+(2*x**4+3*x**3)*ln(2)-x**5+x**4)*ln(( (4+x)*ln(2)-x**2+x)/(ln(2)-x))*ln(ln(((4+x)*ln(2)-x**2+x)/(ln(2)-x)))-x**3 *ln(2)**2+(2*x**4-5*x**3)*ln(2)-x**5)*exp(1/3*x*ln(ln(((4+x)*ln(2)-x**2+x) /(ln(2)-x))))+((3*x**2+9*x-12)*ln(2)**2+(-6*x**3-3*x**2+9*x)*ln(2)+3*x**4- 6*x**3+3*x**2)*exp(x)*ln(((4+x)*ln(2)-x**2+x)/(ln(2)-x)))/((3*x**3+12*x**2 )*ln(2)**2+(-6*x**4-9*x**3)*ln(2)+3*x**5-3*x**4)/ln(((4+x)*ln(2)-x**2+x)/( ln(2)-x)),x)
Time = 0.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=-\frac {x {\left (\log \left (x^{2} - x {\left (\log \left (2\right ) + 1\right )} - 4 \, \log \left (2\right )\right ) - \log \left (x - \log \left (2\right )\right )\right )}^{\frac {1}{3} \, x} - e^{x}}{x} \]
integrate(((((-x^3-4*x^2)*log(2)^2+(2*x^4+3*x^3)*log(2)-x^5+x^4)*log(((4+x )*log(2)-x^2+x)/(log(2)-x))*log(log(((4+x)*log(2)-x^2+x)/(log(2)-x)))-x^3* log(2)^2+(2*x^4-5*x^3)*log(2)-x^5)*exp(1/3*x*log(log(((4+x)*log(2)-x^2+x)/ (log(2)-x))))+((3*x^2+9*x-12)*log(2)^2+(-6*x^3-3*x^2+9*x)*log(2)+3*x^4-6*x ^3+3*x^2)*exp(x)*log(((4+x)*log(2)-x^2+x)/(log(2)-x)))/((3*x^3+12*x^2)*log (2)^2+(-6*x^4-9*x^3)*log(2)+3*x^5-3*x^4)/log(((4+x)*log(2)-x^2+x)/(log(2)- x)),x, algorithm=\
\[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=\int { \frac {3 \, {\left (x^{4} - 2 \, x^{3} + {\left (x^{2} + 3 \, x - 4\right )} \log \left (2\right )^{2} + x^{2} - {\left (2 \, x^{3} + x^{2} - 3 \, x\right )} \log \left (2\right )\right )} e^{x} \log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right ) - {\left (x^{5} + x^{3} \log \left (2\right )^{2} + {\left (x^{5} - x^{4} + {\left (x^{3} + 4 \, x^{2}\right )} \log \left (2\right )^{2} - {\left (2 \, x^{4} + 3 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right ) \log \left (\log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right )\right ) - {\left (2 \, x^{4} - 5 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right )^{\frac {1}{3} \, x}}{3 \, {\left (x^{5} - x^{4} + {\left (x^{3} + 4 \, x^{2}\right )} \log \left (2\right )^{2} - {\left (2 \, x^{4} + 3 \, x^{3}\right )} \log \left (2\right )\right )} \log \left (\frac {x^{2} - {\left (x + 4\right )} \log \left (2\right ) - x}{x - \log \left (2\right )}\right )} \,d x } \]
integrate(((((-x^3-4*x^2)*log(2)^2+(2*x^4+3*x^3)*log(2)-x^5+x^4)*log(((4+x )*log(2)-x^2+x)/(log(2)-x))*log(log(((4+x)*log(2)-x^2+x)/(log(2)-x)))-x^3* log(2)^2+(2*x^4-5*x^3)*log(2)-x^5)*exp(1/3*x*log(log(((4+x)*log(2)-x^2+x)/ (log(2)-x))))+((3*x^2+9*x-12)*log(2)^2+(-6*x^3-3*x^2+9*x)*log(2)+3*x^4-6*x ^3+3*x^2)*exp(x)*log(((4+x)*log(2)-x^2+x)/(log(2)-x)))/((3*x^3+12*x^2)*log (2)^2+(-6*x^4-9*x^3)*log(2)+3*x^5-3*x^4)/log(((4+x)*log(2)-x^2+x)/(log(2)- x)),x, algorithm=\
integrate(1/3*(3*(x^4 - 2*x^3 + (x^2 + 3*x - 4)*log(2)^2 + x^2 - (2*x^3 + x^2 - 3*x)*log(2))*e^x*log((x^2 - (x + 4)*log(2) - x)/(x - log(2))) - (x^5 + x^3*log(2)^2 + (x^5 - x^4 + (x^3 + 4*x^2)*log(2)^2 - (2*x^4 + 3*x^3)*lo g(2))*log((x^2 - (x + 4)*log(2) - x)/(x - log(2)))*log(log((x^2 - (x + 4)* log(2) - x)/(x - log(2)))) - (2*x^4 - 5*x^3)*log(2))*log((x^2 - (x + 4)*lo g(2) - x)/(x - log(2)))^(1/3*x))/((x^5 - x^4 + (x^3 + 4*x^2)*log(2)^2 - (2 *x^4 + 3*x^3)*log(2))*log((x^2 - (x + 4)*log(2) - x)/(x - log(2)))), x)
Time = 10.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {e^x \left (3 x^2-6 x^3+3 x^4+\left (9 x-3 x^2-6 x^3\right ) \log (2)+\left (-12+9 x+3 x^2\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )+\log ^{\frac {x}{3}}\left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \left (-x^5+\left (-5 x^3+2 x^4\right ) \log (2)-x^3 \log ^2(2)+\left (x^4-x^5+\left (3 x^3+2 x^4\right ) \log (2)+\left (-4 x^2-x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right ) \log \left (\log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )\right )\right )}{\left (-3 x^4+3 x^5+\left (-9 x^3-6 x^4\right ) \log (2)+\left (12 x^2+3 x^3\right ) \log ^2(2)\right ) \log \left (\frac {x-x^2+(4+x) \log (2)}{-x+\log (2)}\right )} \, dx=\frac {{\mathrm {e}}^x}{x}-{\mathrm {e}}^{\frac {x\,\ln \left (\ln \left (-\frac {x+4\,\ln \left (2\right )+x\,\ln \left (2\right )-x^2}{x-\ln \left (2\right )}\right )\right )}{3}} \]
int((exp((x*log(log(-(x + log(2)*(x + 4) - x^2)/(x - log(2)))))/3)*(x^3*lo g(2)^2 + log(2)*(5*x^3 - 2*x^4) + x^5 - log(-(x + log(2)*(x + 4) - x^2)/(x - log(2)))*log(log(-(x + log(2)*(x + 4) - x^2)/(x - log(2))))*(log(2)*(3* x^3 + 2*x^4) - log(2)^2*(4*x^2 + x^3) + x^4 - x^5)) - exp(x)*log(-(x + log (2)*(x + 4) - x^2)/(x - log(2)))*(log(2)^2*(9*x + 3*x^2 - 12) - log(2)*(3* x^2 - 9*x + 6*x^3) + 3*x^2 - 6*x^3 + 3*x^4))/(log(-(x + log(2)*(x + 4) - x ^2)/(x - log(2)))*(log(2)*(9*x^3 + 6*x^4) + 3*x^4 - 3*x^5 - log(2)^2*(12*x ^2 + 3*x^3))),x)