3.103.55 \(\int e^{-e^4} (e^{4+e^4-x} (3-3 x)+e^x (1+x)) \, dx\)

Optimal. Leaf size=22 \[ 3 e^{4-x} x+e^{-e^4+x} x \]

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Rubi [B]  time = 0.04, antiderivative size = 48, normalized size of antiderivative = 2.18, number of steps used = 6, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {12, 2176, 2194} \begin {gather*} -3 e^{4-x} (1-x)+3 e^{4-x}-e^{x-e^4}+e^{x-e^4} (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4 + E^4 - x)*(3 - 3*x) + E^x*(1 + x))/E^E^4,x]

[Out]

3*E^(4 - x) - E^(-E^4 + x) - 3*E^(4 - x)*(1 - x) + E^(-E^4 + x)*(1 + x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{-e^4} \int \left (e^{4+e^4-x} (3-3 x)+e^x (1+x)\right ) \, dx\\ &=e^{-e^4} \int e^{4+e^4-x} (3-3 x) \, dx+e^{-e^4} \int e^x (1+x) \, dx\\ &=-3 e^{4-x} (1-x)+e^{-e^4+x} (1+x)-e^{-e^4} \int e^x \, dx-\left (3 e^{-e^4}\right ) \int e^{4+e^4-x} \, dx\\ &=3 e^{4-x}-e^{-e^4+x}-3 e^{4-x} (1-x)+e^{-e^4+x} (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 1.00 \begin {gather*} 3 e^{4-x} x+e^{-e^4+x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4 + E^4 - x)*(3 - 3*x) + E^x*(1 + x))/E^E^4,x]

[Out]

3*E^(4 - x)*x + E^(-E^4 + x)*x

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fricas [A]  time = 0.95, size = 30, normalized size = 1.36 \begin {gather*} {\left (3 \, x e^{\left (-2 \, x + 2 \, e^{4} + 12\right )} + x e^{\left (e^{4} + 8\right )}\right )} e^{\left (x - 2 \, e^{4} - 8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+3)*exp(4)*exp(exp(4)-x)+(x+1)*exp(x))/exp(x)/exp(exp(4)-x),x, algorithm="fricas")

[Out]

(3*x*e^(-2*x + 2*e^4 + 12) + x*e^(e^4 + 8))*e^(x - 2*e^4 - 8)

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giac [A]  time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} {\left (x e^{x} + 3 \, x e^{\left (-x + e^{4} + 4\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+3)*exp(4)*exp(exp(4)-x)+(x+1)*exp(x))/exp(x)/exp(exp(4)-x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 20, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{x -{\mathrm e}^{4}} x +3 x \,{\mathrm e}^{-x +4}\)	\(20\)
norman	\(\left (x \,{\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{2 x}+3 x \,{\mathrm e}^{4}\right ) {\mathrm e}^{-x}\)	\(23\)
default	\({\mathrm e}^{-{\mathrm e}^{4}} {\mathrm e}^{x}+{\mathrm e}^{-{\mathrm e}^{4}} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )-3 \,{\mathrm e}^{-x} {\mathrm e}^{4}-3 \,{\mathrm e}^{4} \left (-x \,{\mathrm e}^{-x}-{\mathrm e}^{-x}\right )\)	\(51\)
meijerg	\(-\frac {3 \,{\mathrm e}^{-x +4-x \,{\mathrm e}^{-{\mathrm e}^{4}}} {\mathrm e}^{2 x} \left (1-\frac {\left (2+2 x \left (2-{\mathrm e}^{-{\mathrm e}^{4}}\right )\right ) {\mathrm e}^{-x \left (2-{\mathrm e}^{-{\mathrm e}^{4}}\right )}}{2}\right )}{\left (2-{\mathrm e}^{-{\mathrm e}^{4}}\right )^{2}}+\frac {3 \,{\mathrm e}^{-x +4-x \,{\mathrm e}^{-{\mathrm e}^{4}}} {\mathrm e}^{2 x} \left (1-{\mathrm e}^{-x \left (2-{\mathrm e}^{-{\mathrm e}^{4}}\right )}\right )}{2-{\mathrm e}^{-{\mathrm e}^{4}}}+{\mathrm e}^{-x \,{\mathrm e}^{-{\mathrm e}^{4}}-x +{\mathrm e}^{4}} {\mathrm e}^{2 x} \left (1-\frac {\left (-2 x \,{\mathrm e}^{-{\mathrm e}^{4}}+2\right ) {\mathrm e}^{x \,{\mathrm e}^{-{\mathrm e}^{4}}}}{2}\right )-{\mathrm e}^{-x \,{\mathrm e}^{-{\mathrm e}^{4}}-x} {\mathrm e}^{2 x} \left (1-{\mathrm e}^{x \,{\mathrm e}^{-{\mathrm e}^{4}}}\right )\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x+3)*exp(4)*exp(exp(4)-x)+(x+1)*exp(x))/exp(x)/exp(exp(4)-x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 1.27, size = 40, normalized size = 1.82 \begin {gather*} {\left (3 \, {\left (x e^{\left (e^{4} + 4\right )} + e^{\left (e^{4} + 4\right )}\right )} e^{\left (-x\right )} + x e^{x} - 3 \, e^{\left (-x + e^{4} + 4\right )}\right )} e^{\left (-e^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+3)*exp(4)*exp(exp(4)-x)+(x+1)*exp(x))/exp(x)/exp(exp(4)-x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.08, size = 19, normalized size = 0.86 \begin {gather*} 3\,x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^4+x\,{\mathrm {e}}^{-{\mathrm {e}}^4}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x - exp(4))*exp(-x)*(exp(x)*(x + 1) - exp(exp(4) - x)*exp(4)*(3*x - 3)),x)

[Out]

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

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sympy [A]  time = 0.20, size = 24, normalized size = 1.09 \begin {gather*} \frac {x e^{x} + 3 x e^{4} e^{- x} e^{e^{4}}}{e^{e^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x+3)*exp(4)*exp(exp(4)-x)+(x+1)*exp(x))/exp(x)/exp(exp(4)-x),x)

[Out]

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

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