3.26.75 \(\int \frac {e^{-\frac {1}{-6-e+x+e^{e^2} x}} (1+e^{e^2})}{36+e^2+e (12-2 x)-12 x+x^2+e^{2 e^2} x^2+e^{e^2} (-12 x-2 e x+2 x^2)} \, dx\)

Optimal. Leaf size=18 \[ e^{\frac {1}{6+e-x-e^{e^2} x}} \]

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Rubi [A]  time = 0.29, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {6, 12, 6688, 2209} \begin {gather*} e^{\frac {1}{-\left (\left (1+e^{e^2}\right ) x\right )+e+6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^E^2)/(E^(-6 - E + x + E^E^2*x)^(-1)*(36 + E^2 + E*(12 - 2*x) - 12*x + x^2 + E^(2*E^2)*x^2 + E^E^2*(
-12*x - 2*E*x + 2*x^2))),x]

[Out]

E^(6 + E - (1 + E^E^2)*x)^(-1)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 {e^{-\frac {1}{-6-e+x+e^{e^2} x}} \left (1+e^{e^2}\right )}{36+e^2+e (12-2 x)-12 x+\left (1+e^{2 e^2}\right ) x^2+e^{e^2} \left (-12 x-2 e x+2 x^2\right )} \, dx\\ &=\left (1+e^{e^2}\right ) \int \frac {e^{-\frac {1}{-6-e+x+e^{e^2} x}}}{36+e^2+e (12-2 x)-12 x+\left (1+e^{2 e^2}\right ) x^2+e^{e^2} \left (-12 x-2 e x+2 x^2\right )} \, dx\\ &=\left (1+e^{e^2}\right ) \int \frac {e^{\frac {1}{6+e-\left (1+e^{e^2}\right ) x}}}{\left (6+e-\left (1+e^{e^2}\right ) x\right )^2} \, dx\\ &=e^{\frac {1}{6+e-\left (1+e^{e^2}\right ) x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 18, normalized size = 1.00 \begin {gather*} e^{\frac {1}{6+e-x-e^{e^2} x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^E^2)/(E^(-6 - E + x + E^E^2*x)^(-1)*(36 + E^2 + E*(12 - 2*x) - 12*x + x^2 + E^(2*E^2)*x^2 + E
^E^2*(-12*x - 2*E*x + 2*x^2))),x]

[Out]

E^(6 + E - x - E^E^2*x)^(-1)

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fricas [A]  time = 0.71, size = 17, normalized size = 0.94 \begin {gather*} e^{\left (-\frac {1}{x e^{\left (e^{2}\right )} + x - e - 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(exp(2)))*exp(-1/(x*exp(exp(2))-exp(1)+x-6))/(x^2*exp(exp(2))^2+(-2*x*exp(1)+2*x^2-12*x)*exp(e
xp(2))+exp(1)^2+(-2*x+12)*exp(1)+x^2-12*x+36),x, algorithm="fricas")

[Out]

e^(-1/(x*e^(e^2) + x - e - 6))

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giac [B]  time = 0.89, size = 39, normalized size = 2.17 \begin {gather*} \frac {{\left (e^{\left (e^{2}\right )} + 1\right )}^{2} e^{\left (-\frac {1}{x e^{\left (e^{2}\right )} + x - e - 6}\right )}}{e^{\left (2 \, e^{2}\right )} + 2 \, e^{\left (e^{2}\right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(exp(2)))*exp(-1/(x*exp(exp(2))-exp(1)+x-6))/(x^2*exp(exp(2))^2+(-2*x*exp(1)+2*x^2-12*x)*exp(e
xp(2))+exp(1)^2+(-2*x+12)*exp(1)+x^2-12*x+36),x, algorithm="giac")

[Out]

(e^(e^2) + 1)^2*e^(-1/(x*e^(e^2) + x - e - 6))/(e^(2*e^2) + 2*e^(e^2) + 1)

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maple [A]  time = 0.49, size = 17, normalized size = 0.94

method	result	size
gosper	\({\mathrm e}^{\frac {1}{{\mathrm e}-x +6-x \,{\mathrm e}^{{\mathrm e}^{2}}}}\)	\(17\)
risch	\({\mathrm e}^{\frac {1}{{\mathrm e}-x +6-x \,{\mathrm e}^{{\mathrm e}^{2}}}}\)	\(17\)
derivativedivides	\(-\frac {\left (1+{\mathrm e}^{{\mathrm e}^{2}}\right ) {\mathrm e}^{\frac {1}{{\mathrm e}-x +6-x \,{\mathrm e}^{{\mathrm e}^{2}}}}}{-1-{\mathrm e}^{{\mathrm e}^{2}}}\)	\(33\)
default	\(-\frac {\left (1+{\mathrm e}^{{\mathrm e}^{2}}\right ) {\mathrm e}^{\frac {1}{{\mathrm e}-x +6-x \,{\mathrm e}^{{\mathrm e}^{2}}}}}{-1-{\mathrm e}^{{\mathrm e}^{2}}}\)	\(33\)
norman	\(\frac {\left ({\mathrm e}+6\right ) {\mathrm e}^{-\frac {1}{x \,{\mathrm e}^{{\mathrm e}^{2}}-{\mathrm e}+x -6}}+\left (-1-{\mathrm e}^{{\mathrm e}^{2}}\right ) x \,{\mathrm e}^{-\frac {1}{x \,{\mathrm e}^{{\mathrm e}^{2}}-{\mathrm e}+x -6}}}{{\mathrm e}-x +6-x \,{\mathrm e}^{{\mathrm e}^{2}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+exp(exp(2)))*exp(-1/(x*exp(exp(2))-exp(1)+x-6))/(x^2*exp(exp(2))^2+(-2*x*exp(1)+2*x^2-12*x)*exp(exp(2))
+exp(1)^2+(-2*x+12)*exp(1)+x^2-12*x+36),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.36, size = 18, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {1}{x {\left (e^{\left (e^{2}\right )} + 1\right )} - e - 6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(exp(2)))*exp(-1/(x*exp(exp(2))-exp(1)+x-6))/(x^2*exp(exp(2))^2+(-2*x*exp(1)+2*x^2-12*x)*exp(e
xp(2))+exp(1)^2+(-2*x+12)*exp(1)+x^2-12*x+36),x, algorithm="maxima")

[Out]

e^(-1/(x*(e^(e^2) + 1) - e - 6))

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mupad [B]  time = 2.08, size = 15, normalized size = 0.83 \begin {gather*} {\mathrm {e}}^{\frac {1}{\mathrm {e}-x\,\left ({\mathrm {e}}^{{\mathrm {e}}^2}+1\right )+6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1/(x - exp(1) + x*exp(exp(2)) - 6))*(exp(exp(2)) + 1))/(exp(2) - 12*x + x^2*exp(2*exp(2)) - exp(exp(
2))*(12*x + 2*x*exp(1) - 2*x^2) + x^2 - exp(1)*(2*x - 12) + 36),x)

[Out]

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

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sympy [A]  time = 0.51, size = 17, normalized size = 0.94 \begin {gather*} e^{- \frac {1}{x + x e^{e^{2}} - 6 - e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+exp(exp(2)))*exp(-1/(x*exp(exp(2))-exp(1)+x-6))/(x**2*exp(exp(2))**2+(-2*x*exp(1)+2*x**2-12*x)*ex
p(exp(2))+exp(1)**2+(-2*x+12)*exp(1)+x**2-12*x+36),x)

[Out]

exp(-1/(x + x*exp(exp(2)) - 6 - E))

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