3.80.27 \(\int e^{-e^x} (5 e^x+e^{e^x} (8 x^3+10 x^4)) \, dx\)

Optimal. Leaf size=18 \[ -5 e^{-e^x}+2 x^4 (1+x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {6688, 2282, 2194, 43} \begin {gather*} 2 x^5+2 x^4-5 e^{-e^x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(5*E^x + E^E^x*(8*x^3 + 10*x^4))/E^E^x,x]

[Out]

-5/E^E^x + 2*x^4 + 2*x^5

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, 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 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 \left (5 e^{-e^x+x}+2 x^3 (4+5 x)\right ) \, dx\\ &=2 \int x^3 (4+5 x) \, dx+5 \int e^{-e^x+x} \, dx\\ &=2 \int \left (4 x^3+5 x^4\right ) \, dx+5 \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^x\right )\\ &=-5 e^{-e^x}+2 x^4+2 x^5\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 20, normalized size = 1.11 \begin {gather*} -5 e^{-e^x}+2 x^4+2 x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5*E^x + E^E^x*(8*x^3 + 10*x^4))/E^E^x,x]

[Out]

-5/E^E^x + 2*x^4 + 2*x^5

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 20, normalized size = 1.11 \begin {gather*} {\left (2 \, {\left (x^{5} + x^{4}\right )} e^{\left (e^{x}\right )} - 5\right )} e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4+8*x^3)*exp(exp(x))+5*exp(x))/exp(exp(x)),x, algorithm="fricas")

[Out]

(2*(x^5 + x^4)*e^(e^x) - 5)*e^(-e^x)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 18, normalized size = 1.00 \begin {gather*} 2 \, x^{5} + 2 \, x^{4} - 5 \, e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4+8*x^3)*exp(exp(x))+5*exp(x))/exp(exp(x)),x, algorithm="giac")

[Out]

2*x^5 + 2*x^4 - 5*e^(-e^x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 19, normalized size = 1.06

method	result	size
default	\(2 x^{5}+2 x^{4}-5 \,{\mathrm e}^{-{\mathrm e}^{x}}\)	\(19\)
risch	\(2 x^{5}+2 x^{4}-5 \,{\mathrm e}^{-{\mathrm e}^{x}}\)	\(19\)
norman	\(\left (-5+2 x^{4} {\mathrm e}^{{\mathrm e}^{x}}+2 x^{5} {\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^4+8*x^3)*exp(exp(x))+5*exp(x))/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 18, normalized size = 1.00 \begin {gather*} 2 \, x^{5} + 2 \, x^{4} - 5 \, e^{\left (-e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^4+8*x^3)*exp(exp(x))+5*exp(x))/exp(exp(x)),x, algorithm="maxima")

[Out]

2*x^5 + 2*x^4 - 5*e^(-e^x)

________________________________________________________________________________________

mupad [B]  time = 0.13, size = 18, normalized size = 1.00 \begin {gather*} 2\,x^4-5\,{\mathrm {e}}^{-{\mathrm {e}}^x}+2\,x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-exp(x))*(5*exp(x) + exp(exp(x))*(8*x^3 + 10*x^4)),x)

[Out]

2*x^4 - 5*exp(-exp(x)) + 2*x^5

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 15, normalized size = 0.83 \begin {gather*} 2 x^{5} + 2 x^{4} - 5 e^{- e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**4+8*x**3)*exp(exp(x))+5*exp(x))/exp(exp(x)),x)

[Out]

2*x**5 + 2*x**4 - 5*exp(-exp(x))

________________________________________________________________________________________