3.74.11 \(\int e^{-x} (1+e^x (-4-4 e^4 x^3+5 x^4)) \, dx\)

Optimal. Leaf size=27 \[ 3-e^{-x}+\left (-1+\frac {e^4}{x}\right ) x \left (4-x^4\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {6688, 2194} \begin {gather*} x^5-e^4 x^4-4 x-e^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^x*(-4 - 4*E^4*x^3 + 5*x^4))/E^x,x]

[Out]

-E^(-x) - 4*x - E^4*x^4 + x^5

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4+e^{-x}-4 e^4 x^3+5 x^4\right ) \, dx\\ &=-4 x-e^4 x^4+x^5+\int e^{-x} \, dx\\ &=-e^{-x}-4 x-e^4 x^4+x^5\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 22, normalized size = 0.81 \begin {gather*} -e^{-x}-4 x-e^4 x^4+x^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^x*(-4 - 4*E^4*x^3 + 5*x^4))/E^x,x]

[Out]

-E^(-x) - 4*x - E^4*x^4 + x^5

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 24, normalized size = 0.89 \begin {gather*} {\left ({\left (x^{5} - x^{4} e^{4} - 4 \, x\right )} e^{x} - 1\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(4)+5*x^4-4)*exp(x)+1)/exp(x),x, algorithm="fricas")

[Out]

((x^5 - x^4*e^4 - 4*x)*e^x - 1)*e^(-x)

________________________________________________________________________________________

giac [A]  time = 0.24, size = 20, normalized size = 0.74 \begin {gather*} x^{5} - x^{4} e^{4} - 4 \, x - e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(4)+5*x^4-4)*exp(x)+1)/exp(x),x, algorithm="giac")

[Out]

x^5 - x^4*e^4 - 4*x - e^(-x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 21, normalized size = 0.78

method	result	size
default	\(-4 x -{\mathrm e}^{-x}+x^{5}-x^{4} {\mathrm e}^{4}\)	\(21\)
risch	\(-4 x -{\mathrm e}^{-x}+x^{5}-x^{4} {\mathrm e}^{4}\)	\(21\)
norman	\(\left (-1+x^{5} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x -x^{4} {\mathrm e}^{4} {\mathrm e}^{x}\right ) {\mathrm e}^{-x}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3*exp(4)+5*x^4-4)*exp(x)+1)/exp(x),x,method=_RETURNVERBOSE)

[Out]

-4*x-1/exp(x)+x^5-x^4*exp(4)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 20, normalized size = 0.74 \begin {gather*} x^{5} - x^{4} e^{4} - 4 \, x - e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3*exp(4)+5*x^4-4)*exp(x)+1)/exp(x),x, algorithm="maxima")

[Out]

x^5 - x^4*e^4 - 4*x - e^(-x)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 20, normalized size = 0.74 \begin {gather*} x^5-{\mathrm {e}}^{-x}-x^4\,{\mathrm {e}}^4-4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x)*(exp(x)*(4*x^3*exp(4) - 5*x^4 + 4) - 1),x)

[Out]

x^5 - exp(-x) - x^4*exp(4) - 4*x

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 17, normalized size = 0.63 \begin {gather*} x^{5} - x^{4} e^{4} - 4 x - e^{- x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3*exp(4)+5*x**4-4)*exp(x)+1)/exp(x),x)

[Out]

x**5 - x**4*exp(4) - 4*x - exp(-x)

________________________________________________________________________________________