3.48.80 \(\int \frac {-8 e^3+e^{-3+e^{-6+x}+x}}{16 e^3} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{16} e^{e^{-6+x}}-\frac {x}{2} \]

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Rubi [A]  time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {12, 2282, 2194} \begin {gather*} \frac {e^{e^{x-6}}}{16}-\frac {x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-8*E^3 + E^(-3 + E^(-6 + x) + x))/(16*E^3),x]

[Out]

E^E^(-6 + x)/16 - x/2

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-8 e^3+e^{-3+e^{-6+x}+x}\right ) \, dx}{16 e^3}\\ &=-\frac {x}{2}+\frac {\int e^{-3+e^{-6+x}+x} \, dx}{16 e^3}\\ &=-\frac {x}{2}+\frac {\operatorname {Subst}\left (\int e^{-3+\frac {x}{e^6}} \, dx,x,e^x\right )}{16 e^3}\\ &=\frac {1}{16} e^{e^{-6+x}}-\frac {x}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 0.88 \begin {gather*} \frac {1}{16} \left (e^{e^{-6+x}}-8 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8*E^3 + E^(-3 + E^(-6 + x) + x))/(16*E^3),x]

[Out]

(E^E^(-6 + x) - 8*x)/16

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fricas [B]  time = 0.64, size = 26, normalized size = 1.53 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x e^{\left (x - 3\right )} - e^{\left (x + e^{\left (x - 6\right )} - 3\right )}\right )} e^{\left (-x + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(exp(x-3)*exp(exp(x-3)/exp(3))-8*exp(3))/exp(3),x, algorithm="fricas")

[Out]

-1/16*(8*x*e^(x - 3) - e^(x + e^(x - 6) - 3))*e^(-x + 3)

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giac [A]  time = 0.24, size = 19, normalized size = 1.12 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x e^{3} - e^{\left (e^{\left (x - 6\right )} + 3\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(exp(x-3)*exp(exp(x-3)/exp(3))-8*exp(3))/exp(3),x, algorithm="giac")

[Out]

-1/16*(8*x*e^3 - e^(e^(x - 6) + 3))*e^(-3)

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maple [A]  time = 0.13, size = 12, normalized size = 0.71

method	result	size
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{x -6}}}{16}-\frac {x}{2}\)	\(12\)
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{x -3} {\mathrm e}^{-3}}}{16}-\frac {x}{2}\)	\(17\)
default	\(\frac {{\mathrm e}^{-3} \left ({\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x -3} {\mathrm e}^{-3}}-8 x \,{\mathrm e}^{3}\right )}{16}\)	\(26\)
derivativedivides	\(\frac {{\mathrm e}^{-3} \left ({\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x -3} {\mathrm e}^{-3}}-8 \,{\mathrm e}^{3} \ln \left ({\mathrm e}^{x -3} {\mathrm e}^{-3}\right )\right )}{16}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*(exp(x-3)*exp(exp(x-3)/exp(3))-8*exp(3))/exp(3),x,method=_RETURNVERBOSE)

[Out]

1/16*exp(exp(x-6))-1/2*x

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maxima [A]  time = 0.36, size = 19, normalized size = 1.12 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x e^{3} - e^{\left (e^{\left (x - 6\right )} + 3\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(exp(x-3)*exp(exp(x-3)/exp(3))-8*exp(3))/exp(3),x, algorithm="maxima")

[Out]

-1/16*(8*x*e^3 - e^(e^(x - 6) + 3))*e^(-3)

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mupad [B]  time = 0.12, size = 12, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^{-6}\,{\mathrm {e}}^x}}{16}-\frac {x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-3)*(exp(3)/2 - (exp(x - 3)*exp(exp(x - 3)*exp(-3)))/16),x)

[Out]

exp(exp(-6)*exp(x))/16 - x/2

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sympy [A]  time = 0.14, size = 14, normalized size = 0.82 \begin {gather*} - \frac {x}{2} + \frac {e^{\frac {e^{x - 3}}{e^{3}}}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*(exp(x-3)*exp(exp(x-3)/exp(3))-8*exp(3))/exp(3),x)

[Out]

-x/2 + exp(exp(-3)*exp(x - 3))/16

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