3.9.80 \(\int \frac {1}{4} (-1-20 e^{-4+x}+4 e^x) \, dx\)

Optimal. Leaf size=21 \[ -3-5 e^{-4+x}+e^x+\frac {1}{4} (-1-x) \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2194} \begin {gather*} -\frac {x}{4}-5 e^{x-4}+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 20*E^(-4 + x) + 4*E^x)/4,x]

[Out]

-5*E^(-4 + x) + 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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \left (-1-20 e^{-4+x}+4 e^x\right ) \, dx\\ &=-\frac {x}{4}-5 \int e^{-4+x} \, dx+\int e^x \, dx\\ &=-5 e^{-4+x}+e^x-\frac {x}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.76 \begin {gather*} -5 e^{-4+x}+e^x-\frac {x}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 20*E^(-4 + x) + 4*E^x)/4,x]

[Out]

-5*E^(-4 + x) + E^x - x/4

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fricas [A]  time = 0.53, size = 17, normalized size = 0.81 \begin {gather*} -\frac {1}{4} \, {\left (x e^{4} - 4 \, {\left (e^{4} - 5\right )} e^{x}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(x*e^4 - 4*(e^4 - 5)*e^x)*e^(-4)

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giac [A]  time = 0.36, size = 12, normalized size = 0.57 \begin {gather*} -\frac {1}{4} \, x - 5 \, e^{\left (x - 4\right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x - 5*e^(x - 4) + e^x

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maple [A]  time = 0.02, size = 13, normalized size = 0.62

method	result	size
default	\(-\frac {x}{4}-5 \,{\mathrm e}^{x -4}+{\mathrm e}^{x}\)	\(13\)
risch	\(-\frac {x}{4}-5 \,{\mathrm e}^{x -4}+{\mathrm e}^{x}\)	\(13\)
norman	\(\left ({\mathrm e}^{4}-5\right ) {\mathrm e}^{-4} {\mathrm e}^{x}-\frac {x}{4}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 12, normalized size = 0.57 \begin {gather*} -\frac {1}{4} \, x - 5 \, e^{\left (x - 4\right )} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*x - 5*e^(x - 4) + e^x

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mupad [B]  time = 0.07, size = 14, normalized size = 0.67 \begin {gather*} -\frac {x}{4}-{\mathrm {e}}^x\,\left (5\,{\mathrm {e}}^{-4}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x) - 5*exp(x - 4) - 1/4,x)

[Out]

- x/4 - exp(x)*(5*exp(-4) - 1)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.67 \begin {gather*} - \frac {x}{4} + \frac {\left (-5 + e^{4}\right ) e^{x}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/4 + (-5 + exp(4))*exp(-4)*exp(x)

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