\(\int \frac {1}{4} e^{\frac {1}{4} (24-4 e^{6-x}-4 e^x-x)} (-1+4 e^{6-x}-4 e^x) \, dx\) [6353]

Optimal result

Integrand size = 45, antiderivative size = 23 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{6-e^{6-x}-e^x-\frac {x}{4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {12, 6838} \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{\frac {1}{4} \left (-x-4 e^{6-x}-4 e^x+24\right )} \]

[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 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{6-e^{6-x}-e^x-\frac {x}{4}} \]

[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)

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{\left (-\frac {1}{4} \, {\left ({\left (x - 24\right )} e^{x} + 4 \, e^{6} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (-x\right )}\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{- \frac {x}{4} - e^{x} + 6 - e^{6} e^{- x}} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{\left (-\frac {1}{4} \, x - e^{x} - e^{\left (-x + 6\right )} + 6\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx=e^{\left (-\frac {1}{4} \, x - e^{x} - e^{\left (-x + 6\right )} + 6\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{4} e^{\frac {1}{4} \left (24-4 e^{6-x}-4 e^x-x\right )} \left (-1+4 e^{6-x}-4 e^x\right ) \, dx={\mathrm {e}}^{-\frac {x}{4}}\,{\mathrm {e}}^6\,{\mathrm {e}}^{-{\mathrm {e}}^{-x}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{-{\mathrm {e}}^x} \]

[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))