3.94.48 \(\int (e^2-2 e^{3+x}+2 e^{-1+e^{-3+x^2}+x^2} x) \, dx\)

Optimal. Leaf size=23 \[ e^2 \left (5+e^{e^{-3+x^2}}-2 e^{1+x}+x\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2194, 6715, 2282} \begin {gather*} e^{e^{x^2-3}+2}+e^2 x-2 e^{x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^2 - 2*E^(3 + x) + 2*E^(-1 + E^(-3 + x^2) + x^2)*x,x]

[Out]

E^(2 + E^(-3 + x^2)) - 2*E^(3 + x) + E^2*x

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^2 x-2 \int e^{3+x} \, dx+2 \int e^{-1+e^{-3+x^2}+x^2} x \, dx\\ &=-2 e^{3+x}+e^2 x+\operatorname {Subst}\left (\int e^{-1+e^{-3+x}+x} \, dx,x,x^2\right )\\ &=-2 e^{3+x}+e^2 x+\operatorname {Subst}\left (\int e^{-1+\frac {x}{e^3}} \, dx,x,e^{x^2}\right )\\ &=e^{2+e^{-3+x^2}}-2 e^{3+x}+e^2 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 24, normalized size = 1.04 \begin {gather*} e^{2+e^{-3+x^2}}-2 e^{3+x}+e^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^2 - 2*E^(3 + x) + 2*E^(-1 + E^(-3 + x^2) + x^2)*x,x]

[Out]

E^(2 + E^(-3 + x^2)) - 2*E^(3 + x) + E^2*x

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fricas [A]  time = 0.60, size = 40, normalized size = 1.74 \begin {gather*} {\left (x e^{\left (x^{2} - 1\right )} - e^{\left (x^{2} + x + \log \relax (2)\right )} + e^{\left (x^{2} + e^{\left (x^{2} - 3\right )} - 1\right )}\right )} e^{\left (-x^{2} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)*exp(x^2-3)*exp(exp(x^2-3))-exp(2)*exp(log(2)+x+1)+exp(2),x, algorithm="fricas")

[Out]

(x*e^(x^2 - 1) - e^(x^2 + x + log(2)) + e^(x^2 + e^(x^2 - 3) - 1))*e^(-x^2 + 3)

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giac [A]  time = 0.22, size = 22, normalized size = 0.96 \begin {gather*} x e^{2} - e^{\left (x + \log \relax (2) + 3\right )} + e^{\left (e^{\left (x^{2} - 3\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)*exp(x^2-3)*exp(exp(x^2-3))-exp(2)*exp(log(2)+x+1)+exp(2),x, algorithm="giac")

[Out]

x*e^2 - e^(x + log(2) + 3) + e^(e^(x^2 - 3) + 2)

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maple [A]  time = 0.05, size = 21, normalized size = 0.91

method	result	size
risch	\(-2 \,{\mathrm e}^{3+x}+{\mathrm e}^{2+{\mathrm e}^{x^{2}-3}}+{\mathrm e}^{2} x\)	\(21\)
default	\(-{\mathrm e}^{2} {\mathrm e}^{\ln \relax (2)+x +1}+{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x^{2}-3}}+{\mathrm e}^{2} x\)	\(26\)
norman	\(-{\mathrm e}^{2} {\mathrm e}^{\ln \relax (2)+x +1}+{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x^{2}-3}}+{\mathrm e}^{2} x\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*exp(2)*exp(x^2-3)*exp(exp(x^2-3))-exp(2)*exp(ln(2)+x+1)+exp(2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 20, normalized size = 0.87 \begin {gather*} x e^{2} - 2 \, e^{\left (x + 3\right )} + e^{\left (e^{\left (x^{2} - 3\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)*exp(x^2-3)*exp(exp(x^2-3))-exp(2)*exp(log(2)+x+1)+exp(2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 7.44, size = 22, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-3}}+x\,{\mathrm {e}}^2-2\,{\mathrm {e}}^3\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2) - exp(2)*exp(x + log(2) + 1) + 2*x*exp(exp(x^2 - 3))*exp(2)*exp(x^2 - 3),x)

[Out]

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

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sympy [A]  time = 0.21, size = 26, normalized size = 1.13 \begin {gather*} x e^{2} - 2 e^{2} e^{x + 1} + e^{2} e^{e^{x^{2} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2*x*exp(2)*exp(x**2-3)*exp(exp(x**2-3))-exp(2)*exp(ln(2)+x+1)+exp(2),x)

[Out]

x*exp(2) - 2*exp(2)*exp(x + 1) + exp(2)*exp(exp(x**2 - 3))

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