3.100.35 \(\int \frac {e^{-x^5} (e^{2 e^{\frac {e^{-x^5} (1-2 e^{x^5} x)}{x}}+\frac {e^{-x^5} (1-2 e^{x^5} x)}{x}} (-2-10 x^5)+e^{e^{\frac {e^{-x^5} (1-2 e^{x^5} x)}{x}}+\frac {e^{-x^5} (1-2 e^{x^5} x)}{x}} (8+40 x^5))}{x^2} \, dx\)

Optimal. Leaf size=23 \[ \left (4-e^{e^{-2+\frac {e^{-x^5}}{x}}}\right )^2 \]

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Rubi [A]  time = 1.16, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 12, 6686} \begin {gather*} \left (4-e^{e^{\frac {e^{-x^5}}{x}-2}}\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*E^((1 - 2*E^x^5*x)/(E^x^5*x)) + (1 - 2*E^x^5*x)/(E^x^5*x))*(-2 - 10*x^5) + E^(E^((1 - 2*E^x^5*x)/(E^
x^5*x)) + (1 - 2*E^x^5*x)/(E^x^5*x))*(8 + 40*x^5))/(E^x^5*x^2),x]

[Out]

(4 - E^E^(-2 + 1/(E^x^5*x)))^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^{-2+e^{-2+\frac {e^{-x^5}}{x}}+\frac {e^{-x^5}}{x}-x^5} \left (4-e^{e^{-2+\frac {e^{-x^5}}{x}}}\right ) \left (1+5 x^5\right )}{x^2} \, dx\\ &=2 \int \frac {e^{-2+e^{-2+\frac {e^{-x^5}}{x}}+\frac {e^{-x^5}}{x}-x^5} \left (4-e^{e^{-2+\frac {e^{-x^5}}{x}}}\right ) \left (1+5 x^5\right )}{x^2} \, dx\\ &=\left (4-e^{e^{-2+\frac {e^{-x^5}}{x}}}\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 21, normalized size = 0.91 \begin {gather*} \left (-4+e^{e^{-2+\frac {e^{-x^5}}{x}}}\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^((1 - 2*E^x^5*x)/(E^x^5*x)) + (1 - 2*E^x^5*x)/(E^x^5*x))*(-2 - 10*x^5) + E^(E^((1 - 2*E^x^5*
x)/(E^x^5*x)) + (1 - 2*E^x^5*x)/(E^x^5*x))*(8 + 40*x^5))/(E^x^5*x^2),x]

[Out]

(-4 + E^E^(-2 + 1/(E^x^5*x)))^2

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fricas [B]  time = 0.63, size = 141, normalized size = 6.13 \begin {gather*} {\left (e^{\left (\frac {2 \, {\left (x e^{\left (x^{5} - \frac {{\left (2 \, x e^{\left (x^{5}\right )} - 1\right )} e^{\left (-x^{5}\right )}}{x}\right )} - 2 \, x e^{\left (x^{5}\right )} + 1\right )} e^{\left (-x^{5}\right )}}{x}\right )} - 8 \, e^{\left (\frac {{\left (x e^{\left (x^{5} - \frac {{\left (2 \, x e^{\left (x^{5}\right )} - 1\right )} e^{\left (-x^{5}\right )}}{x}\right )} - 2 \, x e^{\left (x^{5}\right )} + 1\right )} e^{\left (-x^{5}\right )}}{x} - \frac {{\left (2 \, x e^{\left (x^{5}\right )} - 1\right )} e^{\left (-x^{5}\right )}}{x}\right )}\right )} e^{\left (\frac {2 \, {\left (2 \, x e^{\left (x^{5}\right )} - 1\right )} e^{\left (-x^{5}\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^5-2)*exp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5)))^2+(40*x^5+8)*e
xp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5))))/x^2/exp(x^5),x, algorithm="fricas")

[Out]

(e^(2*(x*e^(x^5 - (2*x*e^(x^5) - 1)*e^(-x^5)/x) - 2*x*e^(x^5) + 1)*e^(-x^5)/x) - 8*e^((x*e^(x^5 - (2*x*e^(x^5)
- 1)*e^(-x^5)/x) - 2*x*e^(x^5) + 1)*e^(-x^5)/x - (2*x*e^(x^5) - 1)*e^(-x^5)/x))*e^(2*(2*x*e^(x^5) - 1)*e^(-x^
5)/x)

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giac [A]  time = 0.14, size = 33, normalized size = 1.43 \begin {gather*} e^{\left (2 \, e^{\left (\frac {e^{\left (-x^{5}\right )}}{x} - 2\right )}\right )} - 8 \, e^{\left (e^{\left (\frac {e^{\left (-x^{5}\right )}}{x} - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^5-2)*exp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5)))^2+(40*x^5+8)*e
xp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5))))/x^2/exp(x^5),x, algorithm="giac")

[Out]

e^(2*e^(e^(-x^5)/x - 2)) - 8*e^(e^(e^(-x^5)/x - 2))

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maple [B]  time = 0.12, size = 50, normalized size = 2.17

method	result	size
risch	\({\mathrm e}^{2 \,{\mathrm e}^{-\frac {\left (2 x \,{\mathrm e}^{x^{5}}-1\right ) {\mathrm e}^{-x^{5}}}{x}}}-8 \,{\mathrm e}^{{\mathrm e}^{-\frac {\left (2 x \,{\mathrm e}^{x^{5}}-1\right ) {\mathrm e}^{-x^{5}}}{x}}}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^5-2)*exp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5)))^2+(40*x^5+8)*exp((-2
*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5))))/x^2/exp(x^5),x,method=_RETURNVERBOSE)

[Out]

exp(2*exp(-(2*x*exp(x^5)-1)*exp(-x^5)/x))-8*exp(exp(-(2*x*exp(x^5)-1)*exp(-x^5)/x))

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maxima [A]  time = 0.54, size = 33, normalized size = 1.43 \begin {gather*} e^{\left (2 \, e^{\left (\frac {e^{\left (-x^{5}\right )}}{x} - 2\right )}\right )} - 8 \, e^{\left (e^{\left (\frac {e^{\left (-x^{5}\right )}}{x} - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^5-2)*exp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5)))^2+(40*x^5+8)*e
xp((-2*x*exp(x^5)+1)/x/exp(x^5))*exp(exp((-2*x*exp(x^5)+1)/x/exp(x^5))))/x^2/exp(x^5),x, algorithm="maxima")

[Out]

e^(2*e^(e^(-x^5)/x - 2)) - 8*e^(e^(e^(-x^5)/x - 2))

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mupad [B]  time = 7.65, size = 33, normalized size = 1.43 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x^5}}{x}}\,{\mathrm {e}}^{-2}}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x^5}}{x}}\,{\mathrm {e}}^{-2}}-8\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x^5)*(exp(2*exp(-(exp(-x^5)*(2*x*exp(x^5) - 1))/x))*exp(-(exp(-x^5)*(2*x*exp(x^5) - 1))/x)*(10*x^5
+ 2) - exp(exp(-(exp(-x^5)*(2*x*exp(x^5) - 1))/x))*exp(-(exp(-x^5)*(2*x*exp(x^5) - 1))/x)*(40*x^5 + 8)))/x^2,x
)

[Out]

exp(exp(exp(-x^5)/x)*exp(-2))*(exp(exp(exp(-x^5)/x)*exp(-2)) - 8)

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sympy [B]  time = 0.73, size = 42, normalized size = 1.83 \begin {gather*} e^{2 e^{\frac {\left (- 2 x e^{x^{5}} + 1\right ) e^{- x^{5}}}{x}}} - 8 e^{e^{\frac {\left (- 2 x e^{x^{5}} + 1\right ) e^{- x^{5}}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**5-2)*exp((-2*x*exp(x**5)+1)/x/exp(x**5))*exp(exp((-2*x*exp(x**5)+1)/x/exp(x**5)))**2+(40*x*
*5+8)*exp((-2*x*exp(x**5)+1)/x/exp(x**5))*exp(exp((-2*x*exp(x**5)+1)/x/exp(x**5))))/x**2/exp(x**5),x)

[Out]

exp(2*exp((-2*x*exp(x**5) + 1)*exp(-x**5)/x)) - 8*exp(exp((-2*x*exp(x**5) + 1)*exp(-x**5)/x))

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