Integrand size = 125, antiderivative size = 36 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=e^x \left (-\frac {1}{2} e^{2 e^{-1/x}}+\frac {-x+\log \left (\frac {x}{4}\right )}{-2+x}\right ) \]
Time = 5.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {e^x \left (e^{2 e^{-1/x}} (-2+x)+2 x+\log (16)-2 \log (x)\right )}{2 (-2+x)} \]
Integrate[(E^(x^(-1) + x)*(-4*x + 6*x^2 + 4*x^3 - 2*x^4) + E^(2/E^x^(-1) + x)*(-8 + 8*x - 2*x^2 + E^x^(-1)*(-4*x^2 + 4*x^3 - x^4)) + E^(x^(-1) + x)* (-6*x^2 + 2*x^3)*Log[x/4])/(E^x^(-1)*(8*x^2 - 8*x^3 + 2*x^4)),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.
| \(\displaystyle \int \frac {e^{-1/x} \left (e^{x+\frac {1}{x}} \left (2 x^3-6 x^2\right ) \log \left (\frac {x}{4}\right )+e^{x+\frac {1}{x}} \left (-2 x^4+4 x^3+6 x^2-4 x\right )+e^{x+2 e^{-1/x}} \left (-2 x^2+e^{\frac {1}{x}} \left (-x^4+4 x^3-4 x^2\right )+8 x-8\right )\right )}{2 x^4-8 x^3+8 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{-1/x} \left (e^{x+\frac {1}{x}} \left (2 x^3-6 x^2\right ) \log \left (\frac {x}{4}\right )+e^{x+\frac {1}{x}} \left (-2 x^4+4 x^3+6 x^2-4 x\right )+e^{x+2 e^{-1/x}} \left (-2 x^2+e^{\frac {1}{x}} \left (-x^4+4 x^3-4 x^2\right )+8 x-8\right )\right )}{x^2 \left (2 x^2-8 x+8\right )}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 8 \int -\frac {e^{-1/x} \left (2 e^{x+\frac {1}{x}} \left (x^4-2 x^3-3 x^2+2 x\right )+e^{x+2 e^{-1/x}} \left (2 x^2-8 x+e^{\frac {1}{x}} \left (x^4-4 x^3+4 x^2\right )+8\right )+2 e^{x+\frac {1}{x}} \left (3 x^2-x^3\right ) \log \left (\frac {x}{4}\right )\right )}{16 (2-x)^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{-1/x} \left (2 e^{x+\frac {1}{x}} \left (x^4-2 x^3-3 x^2+2 x\right )+e^{x+2 e^{-1/x}} \left (2 x^2-8 x+e^{\frac {1}{x}} \left (x^4-4 x^3+4 x^2\right )+8\right )+2 e^{x+\frac {1}{x}} \left (3 x^2-x^3\right ) \log \left (\frac {x}{4}\right )\right )}{(2-x)^2 x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (-\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 (x-2)}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 x}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 (x-2)^2}+\frac {e^{x-\frac {1}{x}} \left (e^{2 e^{-1/x}+\frac {1}{x}} x^4+2 e^{\frac {1}{x}} x^4-4 e^{2 e^{-1/x}+\frac {1}{x}} x^3-4 e^{\frac {1}{x}} x^3-2 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^3+2 e^{2 e^{-1/x}} x^2+4 e^{2 e^{-1/x}+\frac {1}{x}} x^2-6 e^{\frac {1}{x}} x^2+6 e^{\frac {1}{x}} \log \left (\frac {x}{4}\right ) x^2-8 e^{2 e^{-1/x}} x+4 e^{\frac {1}{x}} x+8 e^{2 e^{-1/x}}\right )}{4 x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{2} \int e^x \left (\frac {2 \left (x^3-2 x^2-3 x+2\right )}{(x-2)^2 x}+e^{2 e^{-1/x}}-\frac {2 (x-3) \log \left (\frac {x}{4}\right )}{(x-2)^2}+\frac {2 e^{2 e^{-1/x}-\frac {1}{x}}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {2 e^x \left (x^3-2 x^2-3 x+2\right )}{(x-2)^2 x}+e^{x+2 e^{-1/x}}-\frac {2 e^x (x-3) \log \left (\frac {x}{4}\right )}{(x-2)^2}+\frac {2 e^{x+2 e^{-1/x}-\frac {1}{x}}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-2 \int \frac {e^{x+2 e^{-1/x}-\frac {1}{x}}}{x^2}dx-\int e^{x+2 e^{-1/x}}dx-2 e^x+\frac {4 e^x}{2-x}-\frac {2 e^x \log \left (\frac {x}{4}\right )}{2-x}\right )\) |
Int[(E^(x^(-1) + x)*(-4*x + 6*x^2 + 4*x^3 - 2*x^4) + E^(2/E^x^(-1) + x)*(- 8 + 8*x - 2*x^2 + E^x^(-1)*(-4*x^2 + 4*x^3 - x^4)) + E^(x^(-1) + x)*(-6*x^ 2 + 2*x^3)*Log[x/4])/(E^x^(-1)*(8*x^2 - 8*x^3 + 2*x^4)),x]
3.28.64.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, 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]
Timed out.
\[\int \frac {\left (\left (\left (-x^{4}+4 x^{3}-4 x^{2}\right ) {\mathrm e}^{\frac {1}{x}}-2 x^{2}+8 x -8\right ) {\mathrm e}^{x} {\mathrm e}^{2 \,{\mathrm e}^{-\frac {1}{x}}}+\left (2 x^{3}-6 x^{2}\right ) {\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x} \ln \left (\frac {x}{4}\right )+\left (-2 x^{4}+4 x^{3}+6 x^{2}-4 x \right ) {\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {1}{x}}}{2 x^{4}-8 x^{3}+8 x^{2}}d x\]
int((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x))+(2*x ^3-6*x^2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/x)*exp( x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x)
int((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x))+(2*x ^3-6*x^2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/x)*exp( x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.86 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\left ({\left (x - 2\right )} e^{\left ({\left (x e^{\frac {1}{x}} + 2\right )} e^{\left (-\frac {1}{x}\right )} + \frac {1}{x}\right )} + 2 \, x e^{\left (\frac {x^{2} + 1}{x}\right )} - 2 \, e^{\left (\frac {x^{2} + 1}{x}\right )} \log \left (\frac {1}{4} \, x\right )\right )} e^{\left (-\frac {1}{x}\right )}}{2 \, {\left (x - 2\right )}} \]
integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) )+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm=\
-1/2*((x - 2)*e^((x*e^(1/x) + 2)*e^(-1/x) + 1/x) + 2*x*e^((x^2 + 1)/x) - 2 *e^((x^2 + 1)/x)*log(1/4*x))*e^(-1/x)/(x - 2)
Timed out. \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=\text {Timed out} \]
integrate((((-x**4+4*x**3-4*x**2)*exp(1/x)-2*x**2+8*x-8)*exp(x)*exp(2/exp( 1/x))+(2*x**3-6*x**2)*exp(1/x)*exp(x)*ln(1/4*x)+(-2*x**4+4*x**3+6*x**2-4*x )*exp(1/x)*exp(x))/(2*x**4-8*x**3+8*x**2)/exp(1/x),x)
Time = 0.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\left (x - 2\right )} e^{\left (x + 2 \, e^{\left (-\frac {1}{x}\right )}\right )} + 2 \, {\left (x + 2 \, \log \left (2\right ) - \log \left (x\right )\right )} e^{x}}{2 \, {\left (x - 2\right )}} \]
integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) )+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {x e^{x} + 2 \, e^{x} \log \left (2\right ) - e^{x} \log \left (x\right )}{x - 2} - \frac {1}{2} \, e^{\left (x + 2 \, e^{\left (-\frac {1}{x}\right )}\right )} \]
integrate((((-x^4+4*x^3-4*x^2)*exp(1/x)-2*x^2+8*x-8)*exp(x)*exp(2/exp(1/x) )+(2*x^3-6*x^2)*exp(1/x)*exp(x)*log(1/4*x)+(-2*x^4+4*x^3+6*x^2-4*x)*exp(1/ x)*exp(x))/(2*x^4-8*x^3+8*x^2)/exp(1/x),x, algorithm=\
Time = 9.85 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {e^{-1/x} \left (e^{\frac {1}{x}+x} \left (-4 x+6 x^2+4 x^3-2 x^4\right )+e^{2 e^{-1/x}+x} \left (-8+8 x-2 x^2+e^{\frac {1}{x}} \left (-4 x^2+4 x^3-x^4\right )\right )+e^{\frac {1}{x}+x} \left (-6 x^2+2 x^3\right ) \log \left (\frac {x}{4}\right )\right )}{8 x^2-8 x^3+2 x^4} \, dx=-\frac {{\mathrm {e}}^{x+2\,{\mathrm {e}}^{-\frac {1}{x}}}}{2}-\frac {x\,{\mathrm {e}}^x}{x-2}-\frac {x\,\ln \left (\frac {x}{4}\right )\,{\mathrm {e}}^x}{2\,x-x^2} \]