\(\int \frac {e^{\frac {e^{4 e^{e^4}} (e^3 x+x^2)}{-e^3-x+3 e^{4 e^{e^4}} x}} (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} (-e^6-2 e^3 x-x^2))}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} (-6 e^3 x-6 x^2)} \, dx\) [586]

Optimal result

Integrand size = 138, antiderivative size = 28 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\frac {x}{-e^{-4 e^{e^4}}+\frac {3 x}{e^3+x}}} \]

[Out]

exp(x/(3*x/(exp(3)+x)-1/exp(exp(exp(4)))^4))

Rubi [A] (verified)

Time = 2.55 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 6820, 6838} \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=\exp \left (-\frac {e^{4 e^{e^4}} x \left (x+e^3\right )}{-3 e^{4 e^{e^4}} x+x+e^3}\right ) \]

[In]

Int[(E^((E^(4*E^E^4)*(E^3*x + x^2))/(-E^3 - x + 3*E^(4*E^E^4)*x))*(3*E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-E^6 - 2*E
^3*x - x^2)))/(E^6 + 2*E^3*x + x^2 + 9*E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-6*E^3*x - 6*x^2)),x]

[Out]

E^(-((E^(4*E^E^4)*x*(E^3 + x))/(E^3 + x - 3*E^(4*E^E^4)*x)))

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 6820

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

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}\right ) \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+\left (1+9 e^{8 e^{e^4}}\right ) x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx \\ & = \int \frac {\exp \left (4 e^{e^4}-\frac {e^{4 e^{e^4}} x \left (e^3+x\right )}{e^3+x-3 e^{4 e^{e^4}} x}\right ) \left (-e^6-2 e^3 x-\left (1-3 e^{4 e^{e^4}}\right ) x^2\right )}{\left (e^3+\left (1-3 e^{4 e^{e^4}}\right ) x\right )^2} \, dx \\ & = \exp \left (-\frac {e^{4 e^{e^4}} x \left (e^3+x\right )}{e^3+x-3 e^{4 e^{e^4}} x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{-\frac {e^{4 e^{e^4}} x \left (e^3+x\right )}{e^3+x-3 e^{4 e^{e^4}} x}} \]

[In]

Integrate[(E^((E^(4*E^E^4)*(E^3*x + x^2))/(-E^3 - x + 3*E^(4*E^E^4)*x))*(3*E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-E^6
- 2*E^3*x - x^2)))/(E^6 + 2*E^3*x + x^2 + 9*E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-6*E^3*x - 6*x^2)),x]

[Out]

E^(-((E^(4*E^E^4)*x*(E^3 + x))/(E^3 + x - 3*E^(4*E^E^4)*x)))

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07

method	result	size
risch	\({\mathrm e}^{-\frac {x \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}}\)	\(30\)
parallelrisch	\({\mathrm e}^{-\frac {x \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}}\)	\(30\)
norman	\(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {\left (x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}-{\mathrm e}^{3}-x}}+\left (-3 \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+1\right ) x \,{\mathrm e}^{\frac {\left (x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}-{\mathrm e}^{3}-x}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}\)	\(103\)

[In]

int((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp(4)))^4)*exp((x*exp(3)+x^2)*exp(exp(exp(4)
))^4/(3*x*exp(exp(exp(4)))^4-exp(3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4+exp(3
)^2+2*x*exp(3)+x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {{\left (x^{2} + x e^{3}\right )} e^{\left (4 \, e^{\left (e^{4}\right )}\right )}}{3 \, x e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - x - e^{3}}\right )} \]

[In]

integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp(4)))^4)*exp((x*exp(3)+x^2)*exp(exp(
exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp(3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4
+exp(3)^2+2*x*exp(3)+x^2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\frac {\left (x^{2} + x e^{3}\right ) e^{4 e^{e^{4}}}}{- x + 3 x e^{4 e^{e^{4}}} - e^{3}}} \]

[In]

integrate((3*x**2*exp(exp(exp(4)))**8+(-exp(3)**2-2*x*exp(3)-x**2)*exp(exp(exp(4)))**4)*exp((x*exp(3)+x**2)*ex
p(exp(exp(4)))**4/(3*x*exp(exp(exp(4)))**4-exp(3)-x))/(9*x**2*exp(exp(exp(4)))**8+(-6*x*exp(3)-6*x**2)*exp(exp
(exp(4)))**4+exp(3)**2+2*x*exp(3)+x**2),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (23) = 46\).

Time = 0.43 (sec) , antiderivative size = 218, normalized size of antiderivative = 7.79 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {3 \, x e^{\left (8 \, e^{\left (e^{4}\right )}\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} - \frac {x e^{\left (4 \, e^{\left (e^{4}\right )}\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 6\right )}}{x {\left (27 \, e^{\left (12 \, e^{\left (e^{4}\right )}\right )} - 27 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} + 9 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - 1\right )} - e^{3} - 9 \, e^{\left (8 \, e^{\left (e^{4}\right )} + 3\right )} + 6 \, e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 6\right )}}{x {\left (9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1\right )} + e^{3} - 3 \, e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{3 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - 1}\right )} \]

[In]

integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp(4)))^4)*exp((x*exp(3)+x^2)*exp(exp(
exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp(3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4
+exp(3)^2+2*x*exp(3)+x^2),x, algorithm="maxima")

[Out]

e^(3*x*e^(8*e^(e^4))/(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) - x*e^(4*e^(e^4))/(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^
4)) + 1) + e^(4*e^(e^4) + 6)/(x*(27*e^(12*e^(e^4)) - 27*e^(8*e^(e^4)) + 9*e^(4*e^(e^4)) - 1) - e^3 - 9*e^(8*e^
(e^4) + 3) + 6*e^(4*e^(e^4) + 3)) + e^(4*e^(e^4) + 6)/(x*(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) + e^3 - 3*e^(
4*e^(e^4) + 3)) + e^(4*e^(e^4) + 3)/(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) + e^(4*e^(e^4) + 3)/(3*e^(4*e^(e^4
)) - 1))

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {x^{2} e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + x e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{3 \, x e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - x - e^{3}}\right )} \]

[In]

integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp(4)))^4)*exp((x*exp(3)+x^2)*exp(exp(
exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp(3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4
+exp(3)^2+2*x*exp(3)+x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx={\mathrm {e}}^{-\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,x^2+{\mathrm {e}}^3\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,x}{x+{\mathrm {e}}^3-3\,x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \]

[In]

int((exp(-(exp(4*exp(exp(4)))*(x*exp(3) + x^2))/(x + exp(3) - 3*x*exp(4*exp(exp(4)))))*(3*x^2*exp(8*exp(exp(4)
)) - exp(4*exp(exp(4)))*(exp(6) + 2*x*exp(3) + x^2)))/(exp(6) + 2*x*exp(3) - exp(4*exp(exp(4)))*(6*x*exp(3) +
6*x^2) + 9*x^2*exp(8*exp(exp(4))) + x^2),x)

[Out]

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