3.33.85 $\int \frac{1}{2}(-4+e^{e^3-x}+8e^x)dx$

Optimal. Leaf size=22 $$-\frac{1}{2}{e}^{e^3-x}+4e^x-2x$$

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.100, Rules used = {12, 2194} $$\begin{array}{c}-2x-\frac{e^{e^3-x}}{2}+4e^x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-4 + E^(E^3 - x) + 8*E^x)/2,x]

[Out]

-1/2*E^(E^3 - x) + 4*E^x - 2*x

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{2}\int (-4+e^{e^3-x}+8e^x)dx\\ & =-2x+\frac{1}{2}\int e^{e^3-x}dx+4\int e^{x}dx\\ & =-\frac{1}{2}{e}^{e^3-x}+4e^x-2x\end{array}\end{array}$$

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.09 $$\begin{array}{c}\frac{1}{2}(-e^{e^3-x}+8e^x-4x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-4 + E^(E^3 - x) + 8*E^x)/2,x]

[Out]

(-E^(E^3 - x) + 8*E^x - 4*x)/2

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fricas [A]  time = 0.68, size = 34, normalized size = 1.55 $$\begin{array}{c}-\frac{1}{2}(4x{e}^{(-x+e^3)}+e^{(-2x+2e^3)}-8e^{\left(e^3\right)})e^{(x-e^3)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*exp(-x+exp(3))+4*exp(x)-2,x, algorithm="fricas")

[Out]

-1/2*(4*x*e^(-x + e^3) + e^(-2*x + 2*e^3) - 8*e^(e^3))*e^(x - e^3)

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giac [A]  time = 0.19, size = 17, normalized size = 0.77 $$\begin{array}{c}-2x+4e^x-\frac{1}{2}{e}^{(-x+e^3)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*exp(-x+exp(3))+4*exp(x)-2,x, algorithm="giac")

[Out]

-2*x + 4*e^x - 1/2*e^(-x + e^3)

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maple [A]  time = 0.02, size = 18, normalized size = 0.82

method	result	size
default	$-\frac{{\mathrm{e}}^{-x+{\mathrm{e}}^3}}{2}-2x+4{\mathrm{e}}^x$	$18$
risch	$-\frac{{\mathrm{e}}^{-x+{\mathrm{e}}^3}}{2}-2x+4{\mathrm{e}}^x$	$18$
norman	$(4{\mathrm{e}}^{2x}-2{\mathrm{e}}^{x}x-\frac{{\mathrm{e}}^{{\mathrm{e}}^3}}{2}){\mathrm{e}}^{-x}$	$23$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*exp(-x+exp(3))+4*exp(x)-2,x,method=_RETURNVERBOSE)

[Out]

-1/2*exp(-x+exp(3))-2*x+4*exp(x)

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maxima [A]  time = 0.88, size = 17, normalized size = 0.77 $$\begin{array}{c}-2x+4e^x-\frac{1}{2}{e}^{(-x+e^3)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*exp(-x+exp(3))+4*exp(x)-2,x, algorithm="maxima")

[Out]

-2*x + 4*e^x - 1/2*e^(-x + e^3)

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mupad [B]  time = 1.90, size = 17, normalized size = 0.77 $$\begin{array}{c}4{\mathrm{e}}^x-2x-\frac{{\mathrm{e}}^{-x}{\mathrm{e}}^{{\mathrm{e}}^3}}{2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(3) - x)/2 + 4*exp(x) - 2,x)

[Out]

4*exp(x) - 2*x - (exp(-x)*exp(exp(3)))/2

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sympy [A]  time = 0.14, size = 17, normalized size = 0.77 $$\begin{array}{c}-2x+4e^x-\frac{e^{-x}{e}^{e^3}}{2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*exp(-x+exp(3))+4*exp(x)-2,x)

[Out]

-2*x + 4*exp(x) - exp(-x)*exp(exp(3))/2

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