\(\int \frac {e^{-\frac {e^5+5 x}{x}} (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} (2 x^2-2 x^3-6 x^4+4 x^5)+e^{2 e^{-\frac {e^5+5 x}{x}}} (e^5 (4+4 x-4 x^2)+e^{\frac {e^5+5 x}{x}} (2 x^2-4 x^3)))}{x^2} \, dx\) [2897]

Optimal result

Integrand size = 135, antiderivative size = 31 \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=-2+e^4+\left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right )^2 \]

[Out]

(1-x^2+exp(2/exp(5+exp(5)/x))+x)^2-2+exp(4)

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6820, 12, 6818} \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=\left (-x^2+x+e^{2 e^{-\frac {e^5}{x}-5}}+1\right )^2 \]

[In]

Int[(4*E^(5 + 4/E^((E^5 + 5*x)/x)) + E^((E^5 + 5*x)/x)*(2*x^2 - 2*x^3 - 6*x^4 + 4*x^5) + E^(2/E^((E^5 + 5*x)/x
))*(E^5*(4 + 4*x - 4*x^2) + E^((E^5 + 5*x)/x)*(2*x^2 - 4*x^3)))/(E^((E^5 + 5*x)/x)*x^2),x]

[Out]

(1 + E^(2*E^(-5 - E^5/x)) + x - x^2)^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 e^{-\frac {e^5}{x}} \left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right ) \left (2 e^{2 e^{-5-\frac {e^5}{x}}}+e^{\frac {e^5}{x}} \left (x^2-2 x^3\right )\right )}{x^2} \, dx \\ & = 2 \int \frac {e^{-\frac {e^5}{x}} \left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right ) \left (2 e^{2 e^{-5-\frac {e^5}{x}}}+e^{\frac {e^5}{x}} \left (x^2-2 x^3\right )\right )}{x^2} \, dx \\ & = \left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=\left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right )^2 \]

[In]

Integrate[(4*E^(5 + 4/E^((E^5 + 5*x)/x)) + E^((E^5 + 5*x)/x)*(2*x^2 - 2*x^3 - 6*x^4 + 4*x^5) + E^(2/E^((E^5 +
5*x)/x))*(E^5*(4 + 4*x - 4*x^2) + E^((E^5 + 5*x)/x)*(2*x^2 - 4*x^3)))/(E^((E^5 + 5*x)/x)*x^2),x]

[Out]

(1 + E^(2*E^(-5 - E^5/x)) + x - x^2)^2

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71

method	result	size
risch	\(\left (x^{2}-x -1\right )^{2}+{\mathrm e}^{4 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}}+\left (-2 x^{2}+2 x +2\right ) {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}}\)	\(53\)
parallelrisch	\(-\frac {-x^{5}+2 x^{4}+2 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}} x^{3}+x^{3}-2 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}} x^{2}-x \,{\mathrm e}^{4 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}}-2 x^{2}-2 \,{\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}} x}{x}\)	\(107\)

[In]

int((4*exp(5)*exp(2/exp((exp(5)+5*x)/x))^2+((-4*x^3+2*x^2)*exp((exp(5)+5*x)/x)+(-4*x^2+4*x+4)*exp(5))*exp(2/ex
p((exp(5)+5*x)/x))+(4*x^5-6*x^4-2*x^3+2*x^2)*exp((exp(5)+5*x)/x))/x^2/exp((exp(5)+5*x)/x),x,method=_RETURNVERB
OSE)

[Out]

(x^2-x-1)^2+exp(4*exp(-(exp(5)+5*x)/x))+(-2*x^2+2*x+2)*exp(2*exp(-(exp(5)+5*x)/x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=x^{4} - 2 \, x^{3} - x^{2} - 2 \, {\left (x^{2} - x - 1\right )} e^{\left (2 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}\right )} + 2 \, x + e^{\left (4 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}\right )} \]

[In]

integrate((4*exp(5)*exp(2/exp((exp(5)+5*x)/x))^2+((-4*x^3+2*x^2)*exp((exp(5)+5*x)/x)+(-4*x^2+4*x+4)*exp(5))*ex
p(2/exp((exp(5)+5*x)/x))+(4*x^5-6*x^4-2*x^3+2*x^2)*exp((exp(5)+5*x)/x))/x^2/exp((exp(5)+5*x)/x),x, algorithm="
fricas")

[Out]

x^4 - 2*x^3 - x^2 - 2*(x^2 - x - 1)*e^(2*e^(-(5*x + e^5)/x)) + 2*x + e^(4*e^(-(5*x + e^5)/x))

Sympy [A] (verification not implemented)

Time = 3.93 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=x^{4} - 2 x^{3} - x^{2} + 2 x + \left (- 2 x^{2} + 2 x + 2\right ) e^{2 e^{- \frac {5 x + e^{5}}{x}}} + e^{4 e^{- \frac {5 x + e^{5}}{x}}} \]

[In]

integrate((4*exp(5)*exp(2/exp((exp(5)+5*x)/x))**2+((-4*x**3+2*x**2)*exp((exp(5)+5*x)/x)+(-4*x**2+4*x+4)*exp(5)
)*exp(2/exp((exp(5)+5*x)/x))+(4*x**5-6*x**4-2*x**3+2*x**2)*exp((exp(5)+5*x)/x))/x**2/exp((exp(5)+5*x)/x),x)

[Out]

x**4 - 2*x**3 - x**2 + 2*x + (-2*x**2 + 2*x + 2)*exp(2*exp(-(5*x + exp(5))/x)) + exp(4*exp(-(5*x + exp(5))/x))

Maxima [F]

\[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=\int { -\frac {2 \, {\left ({\left (2 \, {\left (x^{2} - x - 1\right )} e^{5} + {\left (2 \, x^{3} - x^{2}\right )} e^{\left (\frac {5 \, x + e^{5}}{x}\right )}\right )} e^{\left (2 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}\right )} - {\left (2 \, x^{5} - 3 \, x^{4} - x^{3} + x^{2}\right )} e^{\left (\frac {5 \, x + e^{5}}{x}\right )} - 2 \, e^{\left (4 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )} + 5\right )}\right )} e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}}{x^{2}} \,d x } \]

[In]

integrate((4*exp(5)*exp(2/exp((exp(5)+5*x)/x))^2+((-4*x^3+2*x^2)*exp((exp(5)+5*x)/x)+(-4*x^2+4*x+4)*exp(5))*ex
p(2/exp((exp(5)+5*x)/x))+(4*x^5-6*x^4-2*x^3+2*x^2)*exp((exp(5)+5*x)/x))/x^2/exp((exp(5)+5*x)/x),x, algorithm="
maxima")

[Out]

x^4 - 2*x^3 - x^2 + 2*x + e^(4*e^(-e^5/x - 5)) - 2*integrate((2*x^2 + (2*x^3 - x^2)*e^(e^5/x) - 2*x - 2)*e^(-e
^5/x + 2*e^(-e^5/x - 5))/x^2, x)

Giac [F]

\[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=\int { -\frac {2 \, {\left ({\left (2 \, {\left (x^{2} - x - 1\right )} e^{5} + {\left (2 \, x^{3} - x^{2}\right )} e^{\left (\frac {5 \, x + e^{5}}{x}\right )}\right )} e^{\left (2 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}\right )} - {\left (2 \, x^{5} - 3 \, x^{4} - x^{3} + x^{2}\right )} e^{\left (\frac {5 \, x + e^{5}}{x}\right )} - 2 \, e^{\left (4 \, e^{\left (-\frac {5 \, x + e^{5}}{x}\right )} + 5\right )}\right )} e^{\left (-\frac {5 \, x + e^{5}}{x}\right )}}{x^{2}} \,d x } \]

[In]

integrate((4*exp(5)*exp(2/exp((exp(5)+5*x)/x))^2+((-4*x^3+2*x^2)*exp((exp(5)+5*x)/x)+(-4*x^2+4*x+4)*exp(5))*ex
p(2/exp((exp(5)+5*x)/x))+(4*x^5-6*x^4-2*x^3+2*x^2)*exp((exp(5)+5*x)/x))/x^2/exp((exp(5)+5*x)/x),x, algorithm="
giac")

[Out]

integrate(-2*((2*(x^2 - x - 1)*e^5 + (2*x^3 - x^2)*e^((5*x + e^5)/x))*e^(2*e^(-(5*x + e^5)/x)) - (2*x^5 - 3*x^
4 - x^3 + x^2)*e^((5*x + e^5)/x) - 2*e^(4*e^(-(5*x + e^5)/x) + 5))*e^(-(5*x + e^5)/x)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-\frac {e^5+5 x}{x}} \left (4 e^{5+4 e^{-\frac {e^5+5 x}{x}}}+e^{\frac {e^5+5 x}{x}} \left (2 x^2-2 x^3-6 x^4+4 x^5\right )+e^{2 e^{-\frac {e^5+5 x}{x}}} \left (e^5 \left (4+4 x-4 x^2\right )+e^{\frac {e^5+5 x}{x}} \left (2 x^2-4 x^3\right )\right )\right )}{x^2} \, dx=\int \frac {{\mathrm {e}}^{-\frac {5\,x+{\mathrm {e}}^5}{x}}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{-\frac {5\,x+{\mathrm {e}}^5}{x}}}\,\left ({\mathrm {e}}^5\,\left (-4\,x^2+4\,x+4\right )+{\mathrm {e}}^{\frac {5\,x+{\mathrm {e}}^5}{x}}\,\left (2\,x^2-4\,x^3\right )\right )+4\,{\mathrm {e}}^5\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-\frac {5\,x+{\mathrm {e}}^5}{x}}}+{\mathrm {e}}^{\frac {5\,x+{\mathrm {e}}^5}{x}}\,\left (4\,x^5-6\,x^4-2\,x^3+2\,x^2\right )\right )}{x^2} \,d x \]

[In]

int((exp(-(5*x + exp(5))/x)*(exp(2*exp(-(5*x + exp(5))/x))*(exp(5)*(4*x - 4*x^2 + 4) + exp((5*x + exp(5))/x)*(
2*x^2 - 4*x^3)) + 4*exp(5)*exp(4*exp(-(5*x + exp(5))/x)) + exp((5*x + exp(5))/x)*(2*x^2 - 2*x^3 - 6*x^4 + 4*x^
5)))/x^2,x)

[Out]

int((exp(-(5*x + exp(5))/x)*(exp(2*exp(-(5*x + exp(5))/x))*(exp(5)*(4*x - 4*x^2 + 4) + exp((5*x + exp(5))/x)*(
2*x^2 - 4*x^3)) + 4*exp(5)*exp(4*exp(-(5*x + exp(5))/x)) + exp((5*x + exp(5))/x)*(2*x^2 - 2*x^3 - 6*x^4 + 4*x^
5)))/x^2, x)