Integrand size = 170, antiderivative size = 35 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {1}{4} e^{e^{4 e^{-x-\frac {2}{x-x^2}} x^2}} \left (1+e^5\right ) \]
Time = 0.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {1}{4} e^{e^{4 e^{\frac {2}{-1+x}-\frac {2}{x}-x} x^2}} \left (1+e^5\right ) \]
Integrate[(E^(E^(E^((2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*x) + E^((2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*x + (2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*(2 - 2*x - 5*x^2 + 4*x^3 - x^4 + E^5*(2 - 2*x - 5*x^2 + 4*x^3 - x^4)))/(4*x - 8*x^2 + 4*x^3),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.
| \(\displaystyle \int \frac {\left (-x^4+4 x^3-5 x^2+e^5 \left (-x^4+4 x^3-5 x^2-2 x+2\right )-2 x+2\right ) \exp \left (\exp \left (x e^{\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}}\right )+x e^{\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}}+\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}\right )}{4 x^3-8 x^2+4 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-x^4+4 x^3-5 x^2+e^5 \left (-x^4+4 x^3-5 x^2-2 x+2\right )-2 x+2\right ) \exp \left (\exp \left (x e^{\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}}\right )+x e^{\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}}+\frac {-x^3+x^2+\left (x^2-x\right ) \log (4 x)+2}{x^2-x}\right )}{x \left (4 x^2-8 x+4\right )}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 16 \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (-x^4+4 x^3-5 x^2-2 x+e^5 \left (-x^4+4 x^3-5 x^2-2 x+2\right )+2\right )}{64 (1-x)^2 x}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (-x^4+4 x^3-5 x^2-2 x+e^5 \left (-x^4+4 x^3-5 x^2-2 x+2\right )+2\right )}{(1-x)^2 x}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{4} \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (-\left (\left (1+e^5\right ) x^4\right )+4 \left (1+e^5\right ) x^3-5 \left (1+e^5\right ) x^2-2 \left (1+e^5\right ) x+2 \left (1+e^5\right )\right )}{(1-x)^2 x}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{4} \int \left (-\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (1+e^5\right ) x-\frac {2 \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (1+e^5\right )}{x-1}-\frac {2 \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (1+e^5\right )}{(x-1)^2}+2 \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (1+e^5\right )+\frac {2 \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) \left (1+e^5\right )}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (2 \left (1+e^5\right ) \int \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right )dx-2 \left (1+e^5\right ) \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right )}{(x-1)^2}dx-2 \left (1+e^5\right ) \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right )}{x-1}dx+2 \left (1+e^5\right ) \int \frac {\exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right )}{x}dx-\left (1+e^5\right ) \int \exp \left (4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2+e^{4 e^{-\frac {-x^3+x^2+2}{x-x^2}} x^2}-\frac {-x^3+x^2-\left (x-x^2\right ) \log (4 x)+2}{x-x^2}\right ) xdx\right )\) |
Int[(E^(E^(E^((2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*x) + E^((2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*x + (2 + x^2 - x^3 + (-x + x^2)*Log[4*x])/(-x + x^2))*(2 - 2*x - 5*x^2 + 4*x^3 - x^4 + E^5*(2 - 2*x - 5*x^2 + 4*x^3 - x^4)))/(4*x - 8*x^2 + 4*x^3),x]
3.11.31.3.1 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[(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])
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(30)=60\).
Time = 0.13 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.46
\[\frac {{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{\frac {x^{2} \ln \left (4 x \right )-x^{3}-x \ln \left (4 x \right )+x^{2}+2}{x \left (-1+x \right )}}}} {\mathrm e}^{5}}{4}+\frac {{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{\frac {x^{2} \ln \left (4 x \right )-x^{3}-x \ln \left (4 x \right )+x^{2}+2}{x \left (-1+x \right )}}}}}{4}\]
int(((-x^4+4*x^3-5*x^2-2*x+2)*exp(5)-x^4+4*x^3-5*x^2-2*x+2)*exp(((x^2-x)*l n(4*x)-x^3+x^2+2)/(x^2-x))*exp(x*exp(((x^2-x)*ln(4*x)-x^3+x^2+2)/(x^2-x))) *exp(exp(x*exp(((x^2-x)*ln(4*x)-x^3+x^2+2)/(x^2-x))))/(4*x^3-8*x^2+4*x),x)
1/4*exp(exp(x*exp((x^2*ln(4*x)-x^3-x*ln(4*x)+x^2+2)/x/(-1+x))))*exp(5)+1/4 *exp(exp(x*exp((x^2*ln(4*x)-x^3-x*ln(4*x)+x^2+2)/x/(-1+x))))
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (30) = 60\).
Time = 0.25 (sec) , antiderivative size = 206, normalized size of antiderivative = 5.89 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {1}{4} \, {\left (e^{5} + 1\right )} e^{\left (-x e^{\left (-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )} - \frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} e^{\left (x e^{\left (-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )}\right )} - {\left (x^{3} - x^{2}\right )} e^{\left (-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x} + \frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )} \]
integrate(((-x^4+4*x^3-5*x^2-2*x+2)*exp(5)-x^4+4*x^3-5*x^2-2*x+2)*exp(((x^ 2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))*exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/( x^2-x)))*exp(exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))))/(4*x^3-8*x^ 2+4*x),x, algorithm=\
1/4*(e^5 + 1)*e^(-x*e^(-(x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x)) - (x^3 - x^2 - (x^2 - x)*e^(x*e^(-(x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x))) - (x^3 - x^2)*e^(-(x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x)) - (x^2 - x)*log(4*x) - 2)/(x^2 - x) + (x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/ (x^2 - x))
Time = 3.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {\left (1 + e^{5}\right ) e^{e^{x e^{\frac {- x^{3} + x^{2} + \left (x^{2} - x\right ) \log {\left (4 x \right )} + 2}{x^{2} - x}}}}}{4} \]
integrate(((-x**4+4*x**3-5*x**2-2*x+2)*exp(5)-x**4+4*x**3-5*x**2-2*x+2)*ex p(((x**2-x)*ln(4*x)-x**3+x**2+2)/(x**2-x))*exp(x*exp(((x**2-x)*ln(4*x)-x** 3+x**2+2)/(x**2-x)))*exp(exp(x*exp(((x**2-x)*ln(4*x)-x**3+x**2+2)/(x**2-x) )))/(4*x**3-8*x**2+4*x),x)
Time = 1.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\frac {1}{4} \, {\left (e^{5} + 1\right )} e^{\left (e^{\left (4 \, x^{2} e^{\left (-x + \frac {2}{x - 1} - \frac {2}{x}\right )}\right )}\right )} \]
integrate(((-x^4+4*x^3-5*x^2-2*x+2)*exp(5)-x^4+4*x^3-5*x^2-2*x+2)*exp(((x^ 2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))*exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/( x^2-x)))*exp(exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))))/(4*x^3-8*x^ 2+4*x),x, algorithm=\
\[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx=\int { -\frac {{\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} + {\left (x^{4} - 4 \, x^{3} + 5 \, x^{2} + 2 \, x - 2\right )} e^{5} + 2 \, x - 2\right )} e^{\left (x e^{\left (-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )} - \frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x} + e^{\left (x e^{\left (-\frac {x^{3} - x^{2} - {\left (x^{2} - x\right )} \log \left (4 \, x\right ) - 2}{x^{2} - x}\right )}\right )}\right )}}{4 \, {\left (x^{3} - 2 \, x^{2} + x\right )}} \,d x } \]
integrate(((-x^4+4*x^3-5*x^2-2*x+2)*exp(5)-x^4+4*x^3-5*x^2-2*x+2)*exp(((x^ 2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))*exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/( x^2-x)))*exp(exp(x*exp(((x^2-x)*log(4*x)-x^3+x^2+2)/(x^2-x))))/(4*x^3-8*x^ 2+4*x),x, algorithm=\
integrate(-1/4*(x^4 - 4*x^3 + 5*x^2 + (x^4 - 4*x^3 + 5*x^2 + 2*x - 2)*e^5 + 2*x - 2)*e^(x*e^(-(x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x)) - (x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x) + e^(x*e^(-(x^3 - x^2 - (x^2 - x)*log(4*x) - 2)/(x^2 - x))))/(x^3 - 2*x^2 + x), x)
Time = 11.97 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.37 \[ \int \frac {e^{e^{e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x}+e^{\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} x+\frac {2+x^2-x^3+\left (-x+x^2\right ) \log (4 x)}{-x+x^2}} \left (2-2 x-5 x^2+4 x^3-x^4+e^5 \left (2-2 x-5 x^2+4 x^3-x^4\right )\right )}{4 x-8 x^2+4 x^3} \, dx={\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x\,x^{\frac {x}{x-x^2}}\,{\mathrm {e}}^{\frac {x^3}{x-x^2}}\,{\mathrm {e}}^{-\frac {x^2}{x-x^2}}\,{\mathrm {e}}^{-\frac {2}{x-x^2}}}{x^{\frac {x^2}{x-x^2}}}}}\,\left (\frac {{\mathrm {e}}^5}{4}+\frac {1}{4}\right ) \]
int(-(exp((log(4*x)*(x - x^2) - x^2 + x^3 - 2)/(x - x^2))*exp(x*exp((log(4 *x)*(x - x^2) - x^2 + x^3 - 2)/(x - x^2)))*exp(exp(x*exp((log(4*x)*(x - x^ 2) - x^2 + x^3 - 2)/(x - x^2))))*(2*x + exp(5)*(2*x + 5*x^2 - 4*x^3 + x^4 - 2) + 5*x^2 - 4*x^3 + x^4 - 2))/(4*x - 8*x^2 + 4*x^3),x)