\(\int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x (-2110 x-4 x^2)+e^{-\frac {1}{e^x-x}} (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2)}{4 e^{2 x}-8 e^x x+4 x^2} \, dx\) [3736]

Optimal result

Integrand size = 97, antiderivative size = 29 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=-2+x+\left (256+2 \left (3+e^{\frac {1}{-e^x+x}}\right )+\frac {x}{4}\right ) (3+x) \]

[Out]

(3+x)*(2*exp(1/(x-exp(x)))+262+1/4*x)-2+x

Rubi [F]

\[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx \]

[In]

Int[(1055*x^2 + 2*x^3 + E^(2*x)*(1055 + 2*x) + E^x*(-2110*x - 4*x^2) + (-24 + 8*E^(2*x) + E^x*(24 - 8*x) - 8*x
+ 8*x^2)/E^(E^x - x)^(-1))/(4*E^(2*x) - 8*E^x*x + 4*x^2),x]

[Out]

(1055*x)/4 + x^2/4 + 2*Defer[Int][E^(-(E^x - x)^(-1)), x] - 6*Defer[Int][1/(E^(E^x - x)^(-1)*(E^x - x)^2), x]
+ 6*Defer[Int][1/(E^(E^x - x)^(-1)*(E^x - x)), x] + 4*Defer[Int][x/(E^(E^x - x)^(-1)*(E^x - x)^2), x] + 2*Defe
r[Int][x/(E^(E^x - x)^(-1)*(E^x - x)), x] + 2*Defer[Int][x^2/(E^(E^x - x)^(-1)*(E^x - x)^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 \left (e^x-x\right )^2} \, dx \\ & = \frac {1}{4} \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{\left (e^x-x\right )^2} \, dx \\ & = \frac {1}{4} \int \left (\frac {8 e^{-\frac {1}{e^x-x}} (3+x)}{e^x-x}+e^{-\frac {1}{e^x-x}} \left (8+1055 e^{\frac {1}{e^x-x}}+2 e^{\frac {1}{e^x-x}} x\right )+\frac {8 e^{-\frac {1}{e^x-x}} \left (-3+2 x+x^2\right )}{\left (e^x-x\right )^2}\right ) \, dx \\ & = \frac {1}{4} \int e^{-\frac {1}{e^x-x}} \left (8+1055 e^{\frac {1}{e^x-x}}+2 e^{\frac {1}{e^x-x}} x\right ) \, dx+2 \int \frac {e^{-\frac {1}{e^x-x}} (3+x)}{e^x-x} \, dx+2 \int \frac {e^{-\frac {1}{e^x-x}} \left (-3+2 x+x^2\right )}{\left (e^x-x\right )^2} \, dx \\ & = \frac {1}{4} \int \left (1055+8 e^{-\frac {1}{e^x-x}}+2 x\right ) \, dx+2 \int \left (\frac {3 e^{-\frac {1}{e^x-x}}}{e^x-x}+\frac {e^{-\frac {1}{e^x-x}} x}{e^x-x}\right ) \, dx+2 \int \left (-\frac {3 e^{-\frac {1}{e^x-x}}}{\left (e^x-x\right )^2}+\frac {2 e^{-\frac {1}{e^x-x}} x}{\left (e^x-x\right )^2}+\frac {e^{-\frac {1}{e^x-x}} x^2}{\left (e^x-x\right )^2}\right ) \, dx \\ & = \frac {1055 x}{4}+\frac {x^2}{4}+2 \int e^{-\frac {1}{e^x-x}} \, dx+2 \int \frac {e^{-\frac {1}{e^x-x}} x}{e^x-x} \, dx+2 \int \frac {e^{-\frac {1}{e^x-x}} x^2}{\left (e^x-x\right )^2} \, dx+4 \int \frac {e^{-\frac {1}{e^x-x}} x}{\left (e^x-x\right )^2} \, dx-6 \int \frac {e^{-\frac {1}{e^x-x}}}{\left (e^x-x\right )^2} \, dx+6 \int \frac {e^{-\frac {1}{e^x-x}}}{e^x-x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\frac {1}{4} \left (1055 x+x^2+e^{-\frac {1}{e^x-x}} (24+8 x)\right ) \]

[In]

Integrate[(1055*x^2 + 2*x^3 + E^(2*x)*(1055 + 2*x) + E^x*(-2110*x - 4*x^2) + (-24 + 8*E^(2*x) + E^x*(24 - 8*x)
- 8*x + 8*x^2)/E^(E^x - x)^(-1))/(4*E^(2*x) - 8*E^x*x + 4*x^2),x]

[Out]

(1055*x + x^2 + (24 + 8*x)/E^(E^x - x)^(-1))/4

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {x^{2}}{4}+\frac {1055 x}{4}+\left (2 x +6\right ) {\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}}\)	\(27\)
parallelrisch	\(\frac {x^{2}}{4}+\frac {1055 x}{4}+2 \,{\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}} x +6 \,{\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}}\)	\(37\)
norman	\(\frac {\frac {1055 x^{2}}{4}+\frac {x^{3}}{4}-\frac {1055 \,{\mathrm e}^{x} x}{4}-\frac {{\mathrm e}^{x} x^{2}}{4}-6 \,{\mathrm e}^{x} {\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}}+6 \,{\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}} x +2 \,{\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}} x^{2}-2 \,{\mathrm e}^{x} {\mathrm e}^{-\frac {1}{{\mathrm e}^{x}-x}} x}{x -{\mathrm e}^{x}}\)	\(94\)

[In]

int(((8*exp(x)^2+(-8*x+24)*exp(x)+8*x^2-8*x-24)*exp(-1/(exp(x)-x))+(2*x+1055)*exp(x)^2+(-4*x^2-2110*x)*exp(x)+
2*x^3+1055*x^2)/(4*exp(x)^2-8*exp(x)*x+4*x^2),x,method=_RETURNVERBOSE)

[Out]

1/4*x^2+1055/4*x+(2*x+6)*exp(-1/(exp(x)-x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\frac {1}{4} \, x^{2} + 2 \, {\left (x + 3\right )} e^{\left (\frac {1}{x - e^{x}}\right )} + \frac {1055}{4} \, x \]

[In]

integrate(((8*exp(x)^2+(-8*x+24)*exp(x)+8*x^2-8*x-24)*exp(-1/(exp(x)-x))+(2*x+1055)*exp(x)^2+(-4*x^2-2110*x)*e
xp(x)+2*x^3+1055*x^2)/(4*exp(x)^2-8*exp(x)*x+4*x^2),x, algorithm="fricas")

[Out]

1/4*x^2 + 2*(x + 3)*e^(1/(x - e^x)) + 1055/4*x

Sympy [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\frac {x^{2}}{4} + \frac {1055 x}{4} + \left (2 x + 6\right ) e^{- \frac {1}{- x + e^{x}}} \]

[In]

integrate(((8*exp(x)**2+(-8*x+24)*exp(x)+8*x**2-8*x-24)*exp(-1/(exp(x)-x))+(2*x+1055)*exp(x)**2+(-4*x**2-2110*
x)*exp(x)+2*x**3+1055*x**2)/(4*exp(x)**2-8*exp(x)*x+4*x**2),x)

[Out]

x**2/4 + 1055*x/4 + (2*x + 6)*exp(-1/(-x + exp(x)))

Maxima [F]

\[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\int { \frac {2 \, x^{3} + 1055 \, x^{2} + {\left (2 \, x + 1055\right )} e^{\left (2 \, x\right )} - 2 \, {\left (2 \, x^{2} + 1055 \, x\right )} e^{x} + 8 \, {\left (x^{2} - {\left (x - 3\right )} e^{x} - x + e^{\left (2 \, x\right )} - 3\right )} e^{\left (\frac {1}{x - e^{x}}\right )}}{4 \, {\left (x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \,d x } \]

[In]

integrate(((8*exp(x)^2+(-8*x+24)*exp(x)+8*x^2-8*x-24)*exp(-1/(exp(x)-x))+(2*x+1055)*exp(x)^2+(-4*x^2-2110*x)*e
xp(x)+2*x^3+1055*x^2)/(4*exp(x)^2-8*exp(x)*x+4*x^2),x, algorithm="maxima")

[Out]

1/4*x^2 + 1055/4*x + 1/4*integrate(8*(x^2 - (x - 3)*e^x - x + e^(2*x) - 3)*e^(1/(x - e^x))/(x^2 - 2*x*e^x + e^
(2*x)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\frac {1}{4} \, {\left (x^{2} e^{x} + 1055 \, x e^{x} + 8 \, x e^{\left (\frac {x^{2} - x e^{x} + 1}{x - e^{x}}\right )} + 24 \, e^{\left (\frac {x^{2} - x e^{x} + 1}{x - e^{x}}\right )}\right )} e^{\left (-x\right )} \]

[In]

integrate(((8*exp(x)^2+(-8*x+24)*exp(x)+8*x^2-8*x-24)*exp(-1/(exp(x)-x))+(2*x+1055)*exp(x)^2+(-4*x^2-2110*x)*e
xp(x)+2*x^3+1055*x^2)/(4*exp(x)^2-8*exp(x)*x+4*x^2),x, algorithm="giac")

[Out]

1/4*(x^2*e^x + 1055*x*e^x + 8*x*e^((x^2 - x*e^x + 1)/(x - e^x)) + 24*e^((x^2 - x*e^x + 1)/(x - e^x)))*e^(-x)

Mupad [B] (verification not implemented)

Time = 8.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {1055 x^2+2 x^3+e^{2 x} (1055+2 x)+e^x \left (-2110 x-4 x^2\right )+e^{-\frac {1}{e^x-x}} \left (-24+8 e^{2 x}+e^x (24-8 x)-8 x+8 x^2\right )}{4 e^{2 x}-8 e^x x+4 x^2} \, dx=\frac {1055\,x}{4}+{\mathrm {e}}^{\frac {1}{x-{\mathrm {e}}^x}}\,\left (2\,x+6\right )+\frac {x^2}{4} \]

[In]

int((exp(2*x)*(2*x + 1055) - exp(1/(x - exp(x)))*(8*x - 8*exp(2*x) + exp(x)*(8*x - 24) - 8*x^2 + 24) - exp(x)*
(2110*x + 4*x^2) + 1055*x^2 + 2*x^3)/(4*exp(2*x) - 8*x*exp(x) + 4*x^2),x)

[Out]

(1055*x)/4 + exp(1/(x - exp(x)))*(2*x + 6) + x^2/4