3.29.88 \(\int \frac {1}{3} (-5+e^{-9-x+e^5 x-e^x x} (-3+3 e^5+e^x (-3-3 x))) \, dx\)

Optimal. Leaf size=25 \[ 1+e^{-9-x+\left (e^5-e^x\right ) x}-\frac {5 x}{3} \]

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Rubi [A]  time = 0.15, antiderivative size = 24, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {12, 6706} \begin {gather*} e^{-e^x x+e^5 x-x-9}-\frac {5 x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + E^(-9 - x + E^5*x - E^x*x)*(-3 + 3*E^5 + E^x*(-3 - 3*x)))/3,x]

[Out]

E^(-9 - x + E^5*x - E^x*x) - (5*x)/3

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int \left (-5+e^{-9-x+e^5 x-e^x x} \left (-3+3 e^5+e^x (-3-3 x)\right )\right ) \, dx\\ &=-\frac {5 x}{3}+\frac {1}{3} \int e^{-9-x+e^5 x-e^x x} \left (-3+3 e^5+e^x (-3-3 x)\right ) \, dx\\ &=e^{-9-x+e^5 x-e^x x}-\frac {5 x}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 27, normalized size = 1.08 \begin {gather*} \frac {1}{3} \left (3 e^{-9-e^x x+\left (-1+e^5\right ) x}-5 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + E^(-9 - x + E^5*x - E^x*x)*(-3 + 3*E^5 + E^x*(-3 - 3*x)))/3,x]

[Out]

(3*E^(-9 - E^x*x + (-1 + E^5)*x) - 5*x)/3

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fricas [A]  time = 0.61, size = 19, normalized size = 0.76 \begin {gather*} -\frac {5}{3} \, x + e^{\left (x e^{5} - x e^{x} - x - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x-3)*exp(x)+3*exp(5)-3)*exp(-exp(x)*x+x*exp(5)-x-9)-5/3,x, algorithm="fricas")

[Out]

-5/3*x + e^(x*e^5 - x*e^x - x - 9)

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giac [A]  time = 0.27, size = 19, normalized size = 0.76 \begin {gather*} -\frac {5}{3} \, x + e^{\left (x e^{5} - x e^{x} - x - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x-3)*exp(x)+3*exp(5)-3)*exp(-exp(x)*x+x*exp(5)-x-9)-5/3,x, algorithm="giac")

[Out]

-5/3*x + e^(x*e^5 - x*e^x - x - 9)

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maple [A]  time = 0.12, size = 20, normalized size = 0.80

method	result	size
default	\(-\frac {5 x}{3}+{\mathrm e}^{-{\mathrm e}^{x} x +x \,{\mathrm e}^{5}-x -9}\)	\(20\)
norman	\(-\frac {5 x}{3}+{\mathrm e}^{-{\mathrm e}^{x} x +x \,{\mathrm e}^{5}-x -9}\)	\(20\)
risch	\(-\frac {5 x}{3}+{\mathrm e}^{-{\mathrm e}^{x} x +x \,{\mathrm e}^{5}-x -9}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((-3*x-3)*exp(x)+3*exp(5)-3)*exp(-exp(x)*x+x*exp(5)-x-9)-5/3,x,method=_RETURNVERBOSE)

[Out]

-5/3*x+exp(-exp(x)*x+x*exp(5)-x-9)

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maxima [A]  time = 0.73, size = 19, normalized size = 0.76 \begin {gather*} -\frac {5}{3} \, x + e^{\left (x e^{5} - x e^{x} - x - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x-3)*exp(x)+3*exp(5)-3)*exp(-exp(x)*x+x*exp(5)-x-9)-5/3,x, algorithm="maxima")

[Out]

-5/3*x + e^(x*e^5 - x*e^x - x - 9)

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mupad [B]  time = 0.13, size = 22, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^{-x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-9}\,{\mathrm {e}}^{x\,{\mathrm {e}}^5}-\frac {5\,x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (exp(x*exp(5) - x - x*exp(x) - 9)*(exp(x)*(3*x + 3) - 3*exp(5) + 3))/3 - 5/3,x)

[Out]

exp(-x*exp(x))*exp(-x)*exp(-9)*exp(x*exp(5)) - (5*x)/3

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sympy [A]  time = 0.19, size = 19, normalized size = 0.76 \begin {gather*} - \frac {5 x}{3} + e^{- x e^{x} - x + x e^{5} - 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((-3*x-3)*exp(x)+3*exp(5)-3)*exp(-exp(x)*x+x*exp(5)-x-9)-5/3,x)

[Out]

-5*x/3 + exp(-x*exp(x) - x + x*exp(5) - 9)

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