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

Optimal. Leaf size=31 \[ -2+e^4+\left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right )^2 \]

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Rubi [A]  time = 0.97, antiderivative size = 26, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 3, integrand size = 135, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6688, 12, 6686} \begin {gather*} \left (-x^2+x+e^{2 e^{-\frac {e^5}{x}-5}}+1\right )^2 \end {gather*}

Antiderivative was successfully verified.

[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 6686

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 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 26, normalized size = 0.84 \begin {gather*} \left (1+e^{2 e^{-5-\frac {e^5}{x}}}+x-x^2\right )^2 \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.59, size = 57, normalized size = 1.84 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [B]  time = 0.08, size = 59, normalized size = 1.90

method	result	size
risch	\(x^{4}-2 x^{3}-x^{2}+{\mathrm e}^{4 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}}+2 x +\left (-2 x^{2}+2 x +2\right ) {\mathrm e}^{2 \,{\mathrm e}^{-\frac {{\mathrm e}^{5}+5 x}{x}}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + e^{\left (4 \, e^{\left (-\frac {e^{5}}{x} - 5\right )}\right )} - 2 \, \int \frac {{\left (2 \, x^{2} + {\left (2 \, x^{3} - x^{2}\right )} e^{\left (\frac {e^{5}}{x}\right )} - 2 \, x - 2\right )} e^{\left (-\frac {e^{5}}{x} + 2 \, e^{\left (-\frac {e^{5}}{x} - 5\right )}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 7.47, size = 51, normalized size = 1.65 \begin {gather*} 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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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