3.36.37 \(\int (4 e^{e^x+x}-2 x) \, dx\)

Optimal. Leaf size=22 \[ -4^{\frac {5}{e^4}}+4 e^{e^x}-x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2282, 2194} \begin {gather*} 4 e^{e^x}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4*E^(E^x + x) - 2*x,x]

[Out]

4*E^E^x - x^2

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]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x^2+4 \int e^{e^x+x} \, dx\\ &=-x^2+4 \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=4 e^{e^x}-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.59 \begin {gather*} 4 e^{e^x}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4*E^(E^x + x) - 2*x,x]

[Out]

4*E^E^x - x^2

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fricas [A]  time = 0.70, size = 20, normalized size = 0.91 \begin {gather*} -{\left (x^{2} e^{x} - 4 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))-2*x,x, algorithm="fricas")

[Out]

-(x^2*e^x - 4*e^(x + e^x))*e^(-x)

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giac [A]  time = 0.13, size = 11, normalized size = 0.50 \begin {gather*} -x^{2} + 4 \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))-2*x,x, algorithm="giac")

[Out]

-x^2 + 4*e^(e^x)

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maple [A]  time = 0.02, size = 12, normalized size = 0.55

method	result	size
default	\(-x^{2}+4 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(12\)
norman	\(-x^{2}+4 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(12\)
risch	\(-x^{2}+4 \,{\mathrm e}^{{\mathrm e}^{x}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(x)*exp(exp(x))-2*x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 11, normalized size = 0.50 \begin {gather*} -x^{2} + 4 \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))-2*x,x, algorithm="maxima")

[Out]

-x^2 + 4*e^(e^x)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.50 \begin {gather*} 4\,{\mathrm {e}}^{{\mathrm {e}}^x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(exp(x))*exp(x) - 2*x,x)

[Out]

4*exp(exp(x)) - x^2

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sympy [A]  time = 0.10, size = 8, normalized size = 0.36 \begin {gather*} - x^{2} + 4 e^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))-2*x,x)

[Out]

-x**2 + 4*exp(exp(x))

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