3.64.53 \(\int \frac {1}{4} e^{\frac {1}{4} (24-4 e^{6-x}-4 e^x-x)} (-1+4 e^{6-x}-4 e^x) \, dx\)

Optimal. Leaf size=23 \[ e^{6-e^{6-x}-e^x-\frac {x}{4}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6706} \begin {gather*} e^{\frac {1}{4} \left (-x-4 e^{6-x}-4 e^x+24\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((24 - 4*E^(6 - x) - 4*E^x - x)/4)*(-1 + 4*E^(6 - x) - 4*E^x))/4,x]

[Out]

E^((24 - 4*E^(6 - x) - 4*E^x - x)/4)

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}{4} \int e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx\\ &=e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 23, normalized size = 1.00 \begin {gather*} e^{6-e^{6-x}-e^x-\frac {x}{4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((24 - 4*E^(6 - x) - 4*E^x - x)/4)*(-1 + 4*E^(6 - x) - 4*E^x))/4,x]

[Out]

E^(6 - E^(6 - x) - E^x - x/4)

________________________________________________________________________________________

fricas [A]  time = 0.80, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left ({\left (x - 24\right )} e^{x} + 4 \, e^{6} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*exp(x)+4*exp(-x+6)-1)*exp(-exp(x)-exp(-x+6)-1/4*x+6),x, algorithm="fricas")

[Out]

e^(-1/4*((x - 24)*e^x + 4*e^6 + 4*e^(2*x))*e^(-x))

________________________________________________________________________________________

giac [A]  time = 0.19, size = 18, normalized size = 0.78 \begin {gather*} e^{\left (-\frac {1}{4} \, x - e^{x} - e^{\left (-x + 6\right )} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*exp(x)+4*exp(-x+6)-1)*exp(-exp(x)-exp(-x+6)-1/4*x+6),x, algorithm="giac")

[Out]

e^(-1/4*x - e^x - e^(-x + 6) + 6)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 19, normalized size = 0.83

method	result	size
norman	\({\mathrm e}^{-{\mathrm e}^{x}-{\mathrm e}^{-x} {\mathrm e}^{6}-\frac {x}{4}+6}\)	\(19\)
risch	\({\mathrm e}^{-{\mathrm e}^{x}-{\mathrm e}^{-x +6}-\frac {x}{4}+6}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-4*exp(x)+4*exp(-x+6)-1)*exp(-exp(x)-exp(-x+6)-1/4*x+6),x,method=_RETURNVERBOSE)

[Out]

exp(-exp(x)-1/exp(x)*exp(6)-1/4*x+6)

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 18, normalized size = 0.78 \begin {gather*} e^{\left (-\frac {1}{4} \, x - e^{x} - e^{\left (-x + 6\right )} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*exp(x)+4*exp(-x+6)-1)*exp(-exp(x)-exp(-x+6)-1/4*x+6),x, algorithm="maxima")

[Out]

e^(-1/4*x - e^x - e^(-x + 6) + 6)

________________________________________________________________________________________

mupad [B]  time = 4.14, size = 21, normalized size = 0.91 \begin {gather*} {\mathrm {e}}^{-\frac {x}{4}}\,{\mathrm {e}}^6\,{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{-{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(6 - exp(6 - x) - exp(x) - x/4)*(4*exp(x) - 4*exp(6 - x) + 1))/4,x)

[Out]

exp(-x/4)*exp(6)*exp(-exp(-x)*exp(6))*exp(-exp(x))

________________________________________________________________________________________

sympy [A]  time = 0.23, size = 15, normalized size = 0.65 \begin {gather*} e^{- \frac {x}{4} - e^{x} + 6 - e^{6} e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-4*exp(x)+4*exp(-x+6)-1)*exp(-exp(x)-exp(-x+6)-1/4*x+6),x)

[Out]

exp(-x/4 - exp(x) + 6 - exp(6)*exp(-x))

________________________________________________________________________________________