3.41.99 $\int \frac{2e^4+e^{\frac{16+32x-9x^2+4x^3-x^4}{4e^4}}(16-9x+6x^2-2x^3)}{2e^4}dx$

Optimal. Leaf size=27 $$e^{\frac{(4-x)x(9+\frac{4}{x}+x^2)}{4e^4}}+x$$

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Rubi [A]  time = 0.11, antiderivative size = 31, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 2, integrand size = 58, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.034, Rules used = {12, 6706} $$\begin{array}{c}{e}^{\frac{-x^4+4x^3-9x^2+32x+16}{4e^4}}+x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(2*E^4 + E^((16 + 32*x - 9*x^2 + 4*x^3 - x^4)/(4*E^4))*(16 - 9*x + 6*x^2 - 2*x^3))/(2*E^4),x]

[Out]

E^((16 + 32*x - 9*x^2 + 4*x^3 - x^4)/(4*E^4)) + 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 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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int (2e^4+e^{\frac{16+32x-9x^2+4x^3-x^4}{4e^4}}(16-9x+6x^2-2x^3))dx}{2e^4}\\ & =x+\frac{\int e^{\frac{16+32x-9x^2+4x^3-x^4}{4e^4}}(16-9x+6x^2-2x^3)dx}{2e^4}\\ & =e^{\frac{16+32x-9x^2+4x^3-x^4}{4e^4}}+x\end{array}\end{array}$$

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Mathematica [A]  time = 0.24, size = 43, normalized size = 1.59 $$\begin{array}{c}{e}^{\frac{4}{e^4}+\frac{8x}{e^4}-\frac{9x^2}{4e^4}+\frac{x^3}{e^4}-\frac{x^4}{4e^4}}+x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(2*E^4 + E^((16 + 32*x - 9*x^2 + 4*x^3 - x^4)/(4*E^4))*(16 - 9*x + 6*x^2 - 2*x^3))/(2*E^4),x]

[Out]

E^(4/E^4 + (8*x)/E^4 - (9*x^2)/(4*E^4) + x^3/E^4 - x^4/(4*E^4)) + x

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fricas [A]  time = 0.87, size = 25, normalized size = 0.93 $$\begin{array}{c}x+e^{(-\frac{1}{4}(x^4-4x^3+9x^2-32x-16)e^{(-4)})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^3+6*x^2-9*x+16)*exp(1/4*(-x^4+4*x^3-9*x^2+32*x+16)/exp(4))+2*exp(4))/exp(4),x, algorithm=
"fricas")

[Out]

x + e^(-1/4*(x^4 - 4*x^3 + 9*x^2 - 32*x - 16)*e^(-4))

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giac [A]  time = 0.12, size = 40, normalized size = 1.48 $$\begin{array}{c}(x{e}^4+e^{(-\frac{1}{4}{x}^4{e}^{(-4)}+x^3{e}^{(-4)}-\frac{9}{4}{x}^2{e}^{(-4)}+8x{e}^{(-4)}+4e^{(-4)}+4)})e^{(-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^3+6*x^2-9*x+16)*exp(1/4*(-x^4+4*x^3-9*x^2+32*x+16)/exp(4))+2*exp(4))/exp(4),x, algorithm=
"giac")

[Out]

(x*e^4 + e^(-1/4*x^4*e^(-4) + x^3*e^(-4) - 9/4*x^2*e^(-4) + 8*x*e^(-4) + 4*e^(-4) + 4))*e^(-4)

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maple [A]  time = 0.09, size = 19, normalized size = 0.70

method	result	size
risch	$x+{\mathrm{e}}^{-\frac{(x-4)(x^3+9x+4){\mathrm{e}}^{-4}}{4}}$	$19$
norman	$x+{\mathrm{e}}^{\frac{(-x^4+4x^3-9x^2+32x+16){\mathrm{e}}^{-4}}{4}}$	$30$
default	$\frac{{\mathrm{e}}^{-4}(2{\mathrm{e}}^4{\mathrm{e}}^{\frac{(-x^4+4x^3-9x^2+32x+16){\mathrm{e}}^{-4}}{4}}+2x{\mathrm{e}}^4)}{2}$	$44$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-2*x^3+6*x^2-9*x+16)*exp(1/4*(-x^4+4*x^3-9*x^2+32*x+16)/exp(4))+2*exp(4))/exp(4),x,method=_RETURNVER
BOSE)

[Out]

x+exp(-1/4*(x-4)*(x^3+9*x+4)*exp(-4))

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maxima [A]  time = 0.53, size = 40, normalized size = 1.48 $$\begin{array}{c}(x{e}^4+e^{(-\frac{1}{4}{x}^4{e}^{(-4)}+x^3{e}^{(-4)}-\frac{9}{4}{x}^2{e}^{(-4)}+8x{e}^{(-4)}+4e^{(-4)}+4)})e^{(-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^3+6*x^2-9*x+16)*exp(1/4*(-x^4+4*x^3-9*x^2+32*x+16)/exp(4))+2*exp(4))/exp(4),x, algorithm=
"maxima")

[Out]

(x*e^4 + e^(-1/4*x^4*e^(-4) + x^3*e^(-4) - 9/4*x^2*e^(-4) + 8*x*e^(-4) + 4*e^(-4) + 4))*e^(-4)

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mupad [B]  time = 0.18, size = 33, normalized size = 1.22 $$\begin{array}{c}x+{\mathrm{e}}^{-\frac{{\mathrm{e}}^{-4}{x}^4}{4}+{\mathrm{e}}^{-4}{x}^3-\frac{9{\mathrm{e}}^{-4}{x}^2}{4}+8{\mathrm{e}}^{-4}x+4{\mathrm{e}}^{-4}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(exp(4) - (exp(exp(-4)*(8*x - (9*x^2)/4 + x^3 - x^4/4 + 4))*(9*x - 6*x^2 + 2*x^3 - 16))/2),x)

[Out]

x + exp(4*exp(-4) + 8*x*exp(-4) - (9*x^2*exp(-4))/4 + x^3*exp(-4) - (x^4*exp(-4))/4)

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sympy [A]  time = 0.17, size = 26, normalized size = 0.96 $$\begin{array}{c}x+e^{\frac{-\frac{x^4}{4}+x^3-\frac{9x^2}{4}+8x+4}{e^4}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x**3+6*x**2-9*x+16)*exp(1/4*(-x**4+4*x**3-9*x**2+32*x+16)/exp(4))+2*exp(4))/exp(4),x)

[Out]

x + exp((-x**4/4 + x**3 - 9*x**2/4 + 8*x + 4)*exp(-4))

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