3.54.52 \(\int \frac {1}{3} (-3+e^{\frac {1}{3} (3+e^{-13-x}-3 x)} (-3-e^{-13-x})) \, dx\)

Optimal. Leaf size=25 \[ e^{1+\frac {1}{3} e^{-3+x-2 (5+x)}-x}-x \]

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Rubi [A]  time = 0.08, antiderivative size = 22, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{3} \left (-3 x+e^{-x-13}+3\right )}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + E^((3 + E^(-13 - x) - 3*x)/3)*(-3 - E^(-13 - x)))/3,x]

[Out]

E^((3 + E^(-13 - x) - 3*x)/3) - 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 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 (-3+e^{\frac {1}{3} \left (3+e^{-13-x}-3 x\right )} \left (-3-e^{-13-x}\right )\right ) \, dx\\ &=-x+\frac {1}{3} \int e^{\frac {1}{3} \left (3+e^{-13-x}-3 x\right )} \left (-3-e^{-13-x}\right ) \, dx\\ &=e^{\frac {1}{3} \left (3+e^{-13-x}-3 x\right )}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 22, normalized size = 0.88 \begin {gather*} e^{1+\frac {e^{-13-x}}{3}-x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + E^((3 + E^(-13 - x) - 3*x)/3)*(-3 - E^(-13 - x)))/3,x]

[Out]

E^(1 + E^(-13 - x)/3 - x) - x

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fricas [A]  time = 1.00, size = 18, normalized size = 0.72 \begin {gather*} -x + e^{\left (-x + \frac {1}{3} \, e^{\left (-x - 13\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(-x-13)-3)*exp(1/3*exp(-x-13)-x+1)-1,x, algorithm="fricas")

[Out]

-x + e^(-x + 1/3*e^(-x - 13) + 1)

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giac [A]  time = 0.14, size = 18, normalized size = 0.72 \begin {gather*} -x + e^{\left (-x + \frac {1}{3} \, e^{\left (-x - 13\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(-x-13)-3)*exp(1/3*exp(-x-13)-x+1)-1,x, algorithm="giac")

[Out]

-x + e^(-x + 1/3*e^(-x - 13) + 1)

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maple [A]  time = 0.04, size = 19, normalized size = 0.76

method	result	size
norman	\({\mathrm e}^{\frac {{\mathrm e}^{-x -13}}{3}-x +1}-x\)	\(19\)
risch	\({\mathrm e}^{\frac {{\mathrm e}^{-x -13}}{3}-x +1}-x\)	\(19\)
default	\(-x +3 \,{\mathrm e} \,{\mathrm e}^{\frac {{\mathrm e}^{-13} {\mathrm e}^{-x}}{3}} {\mathrm e}^{13}+3 \,{\mathrm e} \,{\mathrm e}^{13} \left (\frac {{\mathrm e}^{-x} {\mathrm e}^{-13} {\mathrm e}^{\frac {{\mathrm e}^{-13} {\mathrm e}^{-x}}{3}}}{3}-{\mathrm e}^{\frac {{\mathrm e}^{-13} {\mathrm e}^{-x}}{3}}\right )\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-exp(-x-13)-3)*exp(1/3*exp(-x-13)-x+1)-1,x,method=_RETURNVERBOSE)

[Out]

exp(1/3*exp(-x-13)-x+1)-x

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maxima [A]  time = 0.42, size = 18, normalized size = 0.72 \begin {gather*} -x + e^{\left (-x + \frac {1}{3} \, e^{\left (-x - 13\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(-x-13)-3)*exp(1/3*exp(-x-13)-x+1)-1,x, algorithm="maxima")

[Out]

-x + e^(-x + 1/3*e^(-x - 13) + 1)

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mupad [B]  time = 0.10, size = 20, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-13}}{3}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (exp(exp(- x - 13)/3 - x + 1)*(exp(- x - 13) + 3))/3 - 1,x)

[Out]

exp(-x)*exp(1)*exp((exp(-x)*exp(-13))/3) - x

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sympy [A]  time = 0.13, size = 14, normalized size = 0.56 \begin {gather*} - x + e^{- x + \frac {e^{- x - 13}}{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-exp(-x-13)-3)*exp(1/3*exp(-x-13)-x+1)-1,x)

[Out]

-x + exp(-x + exp(-x - 13)/3 + 1)

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