∫ eex (−324ex−360e2x−72e3x−4e4x)+e2ex (−18e2x−18e3x−2e4x)+2187x2+729exx2+81e2xx2+3e3xx2 _____________________________________________________________________________________________________________________________________________________

3.88.65 $\int \frac {e^{e^x} (-324 e^x-360 e^{2 x}-72 e^{3 x}-4 e^{4 x})+e^{2 e^x} (-18 e^{2 x}-18 e^{3 x}-2 e^{4 x})+2187 x^2+729 e^x x^2+81 e^{2 x} x^2+3 e^{3 x} x^2}{729+243 e^x+27 e^{2 x}+e^{3 x}} \, dx$

Optimal. Leaf size=25 \[ -\left (2+\frac {e^{e^x+x}}{9+e^x}\right )^2+x^3 \]

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Rubi [B] time = 1.85, antiderivative size = 66, normalized size of antiderivative = 2.64, number of steps used = 47, number of rules used = 6, integrand size = 119, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.050, Rules used = {6741, 6742, 2282, 2246, 2178, 2177} \begin {gather*} x^3-4 e^{e^x}-e^{2 e^x}+\frac {36 e^{e^x}}{e^x+9}+\frac {18 e^{2 e^x}}{e^x+9}-\frac {81 e^{2 e^x}}{\left (e^x+9\right )^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(E^E^x*(-324*E^x - 360*E^(2*x) - 72*E^(3*x) - 4*E^(4*x)) + E^(2*E^x)*(-18*E^(2*x) - 18*E^(3*x) - 2*E^(4*x)

) + 2187*x^2 + 729*E^x*x^2 + 81*E^(2*x)*x^2 + 3*E^(3*x)*x^2)/(729 + 243*E^x + 27*E^(2*x) + E^(3*x)),x]

[Out]

-4*E^E^x - E^(2*E^x) - (81*E^(2*E^x))/(9 + E^x)^2 + (36*E^E^x)/(9 + E^x) + (18*E^(2*E^x))/(9 + E^x) + x^3

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m

+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*

(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] && !$UseGamma ===

True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral

Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] && !$UseGamma === True

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),

x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,

c, d, e, n, p}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu

nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F

reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x

] && InverseFunctionQ[F[x]]]

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^x} \left (-324 e^x-360 e^{2 x}-72 e^{3 x}-4 e^{4 x}\right )+e^{2 e^x} \left (-18 e^{2 x}-18 e^{3 x}-2 e^{4 x}\right )+2187 x^2+729 e^x x^2+81 e^{2 x} x^2+3 e^{3 x} x^2}{\left (9+e^x\right )^3} \, dx\\ &=\int \left (-2 e^{e^x+x} \left (2+e^{e^x}\right )-\frac {1458 e^{2 e^x}}{\left (9+e^x\right )^3}+\frac {162 e^{e^x} \left (2+11 e^{e^x}\right )}{\left (9+e^x\right )^2}-\frac {72 e^{e^x} \left (5+7 e^{e^x}\right )}{9+e^x}+3 \left (12 e^{e^x}+12 e^{2 e^x}+x^2\right )\right ) \, dx\\ &=-\left (2 \int e^{e^x+x} \left (2+e^{e^x}\right ) \, dx\right )+3 \int \left (12 e^{e^x}+12 e^{2 e^x}+x^2\right ) \, dx-72 \int \frac {e^{e^x} \left (5+7 e^{e^x}\right )}{9+e^x} \, dx+162 \int \frac {e^{e^x} \left (2+11 e^{e^x}\right )}{\left (9+e^x\right )^2} \, dx-1458 \int \frac {e^{2 e^x}}{\left (9+e^x\right )^3} \, dx\\ &=x^3-2 \operatorname {Subst}\left (\int e^x \left (2+e^x\right ) \, dx,x,e^x\right )+36 \int e^{e^x} \, dx+36 \int e^{2 e^x} \, dx-72 \operatorname {Subst}\left (\int \frac {e^x \left (5+7 e^x\right )}{x (9+x)} \, dx,x,e^x\right )+162 \operatorname {Subst}\left (\int \frac {e^x \left (2+11 e^x\right )}{x (9+x)^2} \, dx,x,e^x\right )-1458 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x (9+x)^3} \, dx,x,e^x\right )\\ &=x^3-2 \operatorname {Subst}\left (\int (2+x) \, dx,x,e^{e^x}\right )+36 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )+36 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )-72 \operatorname {Subst}\left (\int \left (\frac {5 e^x}{x (9+x)}+\frac {7 e^{2 x}}{x (9+x)}\right ) \, dx,x,e^x\right )+162 \operatorname {Subst}\left (\int \left (\frac {2 e^x}{x (9+x)^2}+\frac {11 e^{2 x}}{x (9+x)^2}\right ) \, dx,x,e^x\right )-1458 \operatorname {Subst}\left (\int \left (\frac {e^{2 x}}{729 x}-\frac {e^{2 x}}{9 (9+x)^3}-\frac {e^{2 x}}{81 (9+x)^2}-\frac {e^{2 x}}{729 (9+x)}\right ) \, dx,x,e^x\right )\\ &=-4 e^{e^x}-e^{2 e^x}+x^3+36 \text {Ei}\left (e^x\right )+36 \text {Ei}\left (2 e^x\right )-2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )+2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )+18 \operatorname {Subst}\left (\int \frac {e^{2 x}}{(9+x)^2} \, dx,x,e^x\right )+162 \operatorname {Subst}\left (\int \frac {e^{2 x}}{(9+x)^3} \, dx,x,e^x\right )+324 \operatorname {Subst}\left (\int \frac {e^x}{x (9+x)^2} \, dx,x,e^x\right )-360 \operatorname {Subst}\left (\int \frac {e^x}{x (9+x)} \, dx,x,e^x\right )-504 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x (9+x)} \, dx,x,e^x\right )+1782 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x (9+x)^2} \, dx,x,e^x\right )\\ &=-4 e^{e^x}-e^{2 e^x}-\frac {81 e^{2 e^x}}{\left (9+e^x\right )^2}-\frac {18 e^{2 e^x}}{9+e^x}+x^3+36 \text {Ei}\left (e^x\right )+34 \text {Ei}\left (2 e^x\right )+\frac {2 \text {Ei}\left (2 \left (9+e^x\right )\right )}{e^{18}}+36 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )+162 \operatorname {Subst}\left (\int \frac {e^{2 x}}{(9+x)^2} \, dx,x,e^x\right )+324 \operatorname {Subst}\left (\int \left (\frac {e^x}{81 x}-\frac {e^x}{9 (9+x)^2}-\frac {e^x}{81 (9+x)}\right ) \, dx,x,e^x\right )-360 \operatorname {Subst}\left (\int \left (\frac {e^x}{9 x}-\frac {e^x}{9 (9+x)}\right ) \, dx,x,e^x\right )-504 \operatorname {Subst}\left (\int \left (\frac {e^{2 x}}{9 x}-\frac {e^{2 x}}{9 (9+x)}\right ) \, dx,x,e^x\right )+1782 \operatorname {Subst}\left (\int \left (\frac {e^{2 x}}{81 x}-\frac {e^{2 x}}{9 (9+x)^2}-\frac {e^{2 x}}{81 (9+x)}\right ) \, dx,x,e^x\right )\\ &=-4 e^{e^x}-e^{2 e^x}-\frac {81 e^{2 e^x}}{\left (9+e^x\right )^2}-\frac {180 e^{2 e^x}}{9+e^x}+x^3+36 \text {Ei}\left (e^x\right )+34 \text {Ei}\left (2 e^x\right )+\frac {38 \text {Ei}\left (2 \left (9+e^x\right )\right )}{e^{18}}+4 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )-4 \operatorname {Subst}\left (\int \frac {e^x}{9+x} \, dx,x,e^x\right )+22 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )-22 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )-36 \operatorname {Subst}\left (\int \frac {e^x}{(9+x)^2} \, dx,x,e^x\right )-40 \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )+40 \operatorname {Subst}\left (\int \frac {e^x}{9+x} \, dx,x,e^x\right )-56 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )+56 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )-198 \operatorname {Subst}\left (\int \frac {e^{2 x}}{(9+x)^2} \, dx,x,e^x\right )+324 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )\\ &=-4 e^{e^x}-e^{2 e^x}-\frac {81 e^{2 e^x}}{\left (9+e^x\right )^2}+\frac {36 e^{e^x}}{9+e^x}+\frac {18 e^{2 e^x}}{9+e^x}+x^3+\frac {36 \text {Ei}\left (9+e^x\right )}{e^9}+\frac {396 \text {Ei}\left (2 \left (9+e^x\right )\right )}{e^{18}}-36 \operatorname {Subst}\left (\int \frac {e^x}{9+x} \, dx,x,e^x\right )-396 \operatorname {Subst}\left (\int \frac {e^{2 x}}{9+x} \, dx,x,e^x\right )\\ &=-4 e^{e^x}-e^{2 e^x}-\frac {81 e^{2 e^x}}{\left (9+e^x\right )^2}+\frac {36 e^{e^x}}{9+e^x}+\frac {18 e^{2 e^x}}{9+e^x}+x^3\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.10, size = 38, normalized size = 1.52 \begin {gather*} -\frac {e^{2 \left (e^x+x\right )}}{\left (9+e^x\right )^2}-\frac {4 e^{e^x+x}}{9+e^x}+x^3 \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^E^x*(-324*E^x - 360*E^(2*x) - 72*E^(3*x) - 4*E^(4*x)) + E^(2*E^x)*(-18*E^(2*x) - 18*E^(3*x) - 2*E

^(4*x)) + 2187*x^2 + 729*E^x*x^2 + 81*E^(2*x)*x^2 + 3*E^(3*x)*x^2)/(729 + 243*E^x + 27*E^(2*x) + E^(3*x)),x]

[Out]

-(E^(2*(E^x + x))/(9 + E^x)^2) - (4*E^(E^x + x))/(9 + E^x) + x^3

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fricas [B] time = 0.79, size = 59, normalized size = 2.36 \begin {gather*} \frac {x^{3} e^{\left (2 \, x\right )} + 18 \, x^{3} e^{x} + 81 \, x^{3} - 4 \, {\left (e^{\left (2 \, x\right )} + 9 \, e^{x}\right )} e^{\left (e^{x}\right )} - e^{\left (2 \, x + 2 \, e^{x}\right )}}{e^{\left (2 \, x\right )} + 18 \, e^{x} + 81} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)^4-18*exp(x)^3-18*exp(x)^2)*exp(exp(x))^2+(-4*exp(x)^4-72*exp(x)^3-360*exp(x)^2-324*exp(x

))*exp(exp(x))+3*x^2*exp(x)^3+81*exp(x)^2*x^2+729*exp(x)*x^2+2187*x^2)/(exp(x)^3+27*exp(x)^2+243*exp(x)+729),x

, algorithm="fricas")

[Out]

(x^3*e^(2*x) + 18*x^3*e^x + 81*x^3 - 4*(e^(2*x) + 9*e^x)*e^(e^x) - e^(2*x + 2*e^x))/(e^(2*x) + 18*e^x + 81)

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giac [B] time = 0.18, size = 61, normalized size = 2.44 \begin {gather*} \frac {x^{3} e^{\left (2 \, x\right )} + 18 \, x^{3} e^{x} + 81 \, x^{3} - e^{\left (2 \, x + 2 \, e^{x}\right )} - 4 \, e^{\left (2 \, x + e^{x}\right )} - 36 \, e^{\left (x + e^{x}\right )}}{e^{\left (2 \, x\right )} + 18 \, e^{x} + 81} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)^4-18*exp(x)^3-18*exp(x)^2)*exp(exp(x))^2+(-4*exp(x)^4-72*exp(x)^3-360*exp(x)^2-324*exp(x

))*exp(exp(x))+3*x^2*exp(x)^3+81*exp(x)^2*x^2+729*exp(x)*x^2+2187*x^2)/(exp(x)^3+27*exp(x)^2+243*exp(x)+729),x

, algorithm="giac")

[Out]

(x^3*e^(2*x) + 18*x^3*e^x + 81*x^3 - e^(2*x + 2*e^x) - 4*e^(2*x + e^x) - 36*e^(x + e^x))/(e^(2*x) + 18*e^x + 8

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maple [A] time = 0.14, size = 35, normalized size = 1.40


method	result	size

risch	$x^{3}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x}+2 x}}{\left (9+{\mathrm e}^{x}\right )^{2}}-\frac {4 \,{\mathrm e}^{{\mathrm e}^{x}+x}}{9+{\mathrm e}^{x}}$	$35$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(x)^4-18*exp(x)^3-18*exp(x)^2)*exp(exp(x))^2+(-4*exp(x)^4-72*exp(x)^3-360*exp(x)^2-324*exp(x))*exp

(exp(x))+3*x^2*exp(x)^3+81*exp(x)^2*x^2+729*exp(x)*x^2+2187*x^2)/(exp(x)^3+27*exp(x)^2+243*exp(x)+729),x,metho

d=_RETURNVERBOSE)

[Out]

x^3-1/(9+exp(x))^2*exp(2*exp(x)+2*x)-4/(9+exp(x))*exp(exp(x)+x)

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maxima [B] time = 0.39, size = 59, normalized size = 2.36 \begin {gather*} \frac {x^{3} e^{\left (2 \, x\right )} + 18 \, x^{3} e^{x} + 81 \, x^{3} - 4 \, {\left (e^{\left (2 \, x\right )} + 9 \, e^{x}\right )} e^{\left (e^{x}\right )} - e^{\left (2 \, x + 2 \, e^{x}\right )}}{e^{\left (2 \, x\right )} + 18 \, e^{x} + 81} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)^4-18*exp(x)^3-18*exp(x)^2)*exp(exp(x))^2+(-4*exp(x)^4-72*exp(x)^3-360*exp(x)^2-324*exp(x

))*exp(exp(x))+3*x^2*exp(x)^3+81*exp(x)^2*x^2+729*exp(x)*x^2+2187*x^2)/(exp(x)^3+27*exp(x)^2+243*exp(x)+729),x

, algorithm="maxima")

[Out]

(x^3*e^(2*x) + 18*x^3*e^x + 81*x^3 - 4*(e^(2*x) + 9*e^x)*e^(e^x) - e^(2*x + 2*e^x))/(e^(2*x) + 18*e^x + 81)

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mupad [B] time = 5.55, size = 40, normalized size = 1.60 \begin {gather*} x^3-\frac {4\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^x}{{\mathrm {e}}^x+9}-\frac {{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{{\mathrm {e}}^{2\,x}+18\,{\mathrm {e}}^x+81} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((729*x^2*exp(x) - exp(exp(x))*(360*exp(2*x) + 72*exp(3*x) + 4*exp(4*x) + 324*exp(x)) - exp(2*exp(x))*(18*e

xp(2*x) + 18*exp(3*x) + 2*exp(4*x)) + 81*x^2*exp(2*x) + 3*x^2*exp(3*x) + 2187*x^2)/(27*exp(2*x) + exp(3*x) + 2

43*exp(x) + 729),x)

[Out]

x^3 - (4*exp(exp(x))*exp(x))/(exp(x) + 9) - (exp(2*x)*exp(2*exp(x)))/(exp(2*x) + 18*exp(x) + 81)

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sympy [B] time = 0.22, size = 66, normalized size = 2.64 \begin {gather*} x^{3} + \frac {\left (- e^{3 x} - 9 e^{2 x}\right ) e^{2 e^{x}} + \left (- 4 e^{3 x} - 72 e^{2 x} - 324 e^{x}\right ) e^{e^{x}}}{e^{3 x} + 27 e^{2 x} + 243 e^{x} + 729} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)**4-18*exp(x)**3-18*exp(x)**2)*exp(exp(x))**2+(-4*exp(x)**4-72*exp(x)**3-360*exp(x)**2-32

4*exp(x))*exp(exp(x))+3*x**2*exp(x)**3+81*exp(x)**2*x**2+729*exp(x)*x**2+2187*x**2)/(exp(x)**3+27*exp(x)**2+24

3*exp(x)+729),x)

[Out]

x**3 + ((-exp(3*x) - 9*exp(2*x))*exp(2*exp(x)) + (-4*exp(3*x) - 72*exp(2*x) - 324*exp(x))*exp(exp(x)))/(exp(3*

x) + 27*exp(2*x) + 243*exp(x) + 729)

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3.88.65 \(\int \frac {e^{e^x} (-324 e^x-360 e^{2 x}-72 e^{3 x}-4 e^{4 x})+e^{2 e^x} (-18 e^{2 x}-18 e^{3 x}-2 e^{4 x})+2187 x^2+729 e^x x^2+81 e^{2 x} x^2+3 e^{3 x} x^2}{729+243 e^x+27 e^{2 x}+e^{3 x}} \, dx\)