3.97.70 \(\int \frac {1}{4} (e^{\frac {1}{4} (16-e^{e^x}-4 x)} (-4-e^{e^x+x})+e^{e+27 x-2 x^2} (-108+16 x)) \, dx\)

Optimal. Leaf size=30 \[ e^{4-\frac {e^{e^x}}{4}-x}-e^{e+(27-2 x) x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 33, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 6706, 2236} \begin {gather*} e^{\frac {1}{4} \left (-4 x-e^{e^x}+16\right )}-e^{-2 x^2+27 x+e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((16 - E^E^x - 4*x)/4)*(-4 - E^(E^x + x)) + E^(E + 27*x - 2*x^2)*(-108 + 16*x))/4,x]

[Out]

E^((16 - E^E^x - 4*x)/4) - E^(E + 27*x - 2*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 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

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 \left (e^{\frac {1}{4} \left (16-e^{e^x}-4 x\right )} \left (-4-e^{e^x+x}\right )+e^{e+27 x-2 x^2} (-108+16 x)\right ) \, dx\\ &=\frac {1}{4} \int e^{\frac {1}{4} \left (16-e^{e^x}-4 x\right )} \left (-4-e^{e^x+x}\right ) \, dx+\frac {1}{4} \int e^{e+27 x-2 x^2} (-108+16 x) \, dx\\ &=e^{\frac {1}{4} \left (16-e^{e^x}-4 x\right )}-e^{e+27 x-2 x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 34, normalized size = 1.13 \begin {gather*} e^{\frac {1}{4} \left (16-e^{e^x}\right )-x}-e^{e+27 x-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16 - E^E^x - 4*x)/4)*(-4 - E^(E^x + x)) + E^(E + 27*x - 2*x^2)*(-108 + 16*x))/4,x]

[Out]

E^((16 - E^E^x)/4 - x) - E^(E + 27*x - 2*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 35, normalized size = 1.17 \begin {gather*} -e^{\left (-2 \, x^{2} + 27 \, x + e\right )} + e^{\left (-\frac {1}{4} \, {\left (4 \, {\left (x - 4\right )} e^{x} + e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(x)*exp(exp(x))-4)*exp(-1/4*exp(exp(x))-x+4)+1/4*(16*x-108)*exp(exp(1)-2*x^2+27*x),x, algor
ithm="fricas")

[Out]

-e^(-2*x^2 + 27*x + e) + e^(-1/4*(4*(x - 4)*e^x + e^(x + e^x))*e^(-x))

________________________________________________________________________________________

giac [A]  time = 0.18, size = 26, normalized size = 0.87 \begin {gather*} -e^{\left (-2 \, x^{2} + 27 \, x + e\right )} + e^{\left (-x - \frac {1}{4} \, e^{\left (e^{x}\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(x)*exp(exp(x))-4)*exp(-1/4*exp(exp(x))-x+4)+1/4*(16*x-108)*exp(exp(1)-2*x^2+27*x),x, algor
ithm="giac")

[Out]

-e^(-2*x^2 + 27*x + e) + e^(-x - 1/4*e^(e^x) + 4)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 27, normalized size = 0.90

method	result	size
default	\(-{\mathrm e}^{{\mathrm e}-2 x^{2}+27 x}+{\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4}-x +4}\)	\(27\)
risch	\(-{\mathrm e}^{{\mathrm e}-2 x^{2}+27 x}+{\mathrm e}^{-\frac {{\mathrm e}^{{\mathrm e}^{x}}}{4}-x +4}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-exp(x)*exp(exp(x))-4)*exp(-1/4*exp(exp(x))-x+4)+1/4*(16*x-108)*exp(exp(1)-2*x^2+27*x),x,method=_RETU
RNVERBOSE)

[Out]

-exp(exp(1)-2*x^2+27*x)+exp(-1/4*exp(exp(x))-x+4)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 26, normalized size = 0.87 \begin {gather*} -e^{\left (-2 \, x^{2} + 27 \, x + e\right )} + e^{\left (-x - \frac {1}{4} \, e^{\left (e^{x}\right )} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(x)*exp(exp(x))-4)*exp(-1/4*exp(exp(x))-x+4)+1/4*(16*x-108)*exp(exp(1)-2*x^2+27*x),x, algor
ithm="maxima")

[Out]

-e^(-2*x^2 + 27*x + e) + e^(-x - 1/4*e^(e^x) + 4)

________________________________________________________________________________________

mupad [B]  time = 7.72, size = 26, normalized size = 0.87 \begin {gather*} {\mathrm {e}}^{4-\frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{4}-x}-{\mathrm {e}}^{-2\,x^2+27\,x+\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(27*x + exp(1) - 2*x^2)*(16*x - 108))/4 - (exp(4 - exp(exp(x))/4 - x)*(exp(exp(x))*exp(x) + 4))/4,x)

[Out]

exp(4 - exp(exp(x))/4 - x) - exp(27*x + exp(1) - 2*x^2)

________________________________________________________________________________________

sympy [A]  time = 0.40, size = 24, normalized size = 0.80 \begin {gather*} e^{- x - \frac {e^{e^{x}}}{4} + 4} - e^{- 2 x^{2} + 27 x + e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-exp(x)*exp(exp(x))-4)*exp(-1/4*exp(exp(x))-x+4)+1/4*(16*x-108)*exp(exp(1)-2*x**2+27*x),x)

[Out]

exp(-x - exp(exp(x))/4 + 4) - exp(-2*x**2 + 27*x + E)

________________________________________________________________________________________