3.41.67 $\int \frac{1}{9}{e}^{-24-x}(27e^{24+4x}+e^{24}(-9+9x)+e^x(32x^7-256e^3{x}^7+896e^6{x}^7-1792e^9{x}^7+2240e^{12}{x}^7-1792e^{15}{x}^7+896e^{18}{x}^7-256e^{21}{x}^7+32e^{24}{x}^7))dx$

Optimal. Leaf size=31 $$-e^{-x}(-e^{4x}+x)+\frac{4}{9}{(x-\frac{x}{e^3})}^8$$

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Rubi [A]  time = 0.19, antiderivative size = 43, normalized size of antiderivative = 1.39, number of steps used = 6, number of rules used = 4, integrand size = 104, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.038, Rules used = {12, 6742, 2194, 2176} $$\begin{array}{c}\frac{4{(1-e^3)}^8{x}^8}{9e^{24}}-e^{-x}+e^{3x}+e^{-x}(1-x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^3*x^7 + 896*E^6*x^7 - 1792*E^9*x^7 +
2240*E^12*x^7 - 1792*E^15*x^7 + 896*E^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]

[Out]

-E^(-x) + E^(3*x) + (1 - x)/E^x + (4*(1 - E^3)^8*x^8)/(9*E^24)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 6742

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

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{9}\int e^{-24-x}(27e^{24+4x}+e^{24}(-9+9x)+e^x(32x^7-256e^3{x}^7+896e^6{x}^7-1792e^9{x}^7+2240e^{12}{x}^7-1792e^{15}{x}^7+896e^{18}{x}^7-256e^{21}{x}^7+32e^{24}{x}^7))dx\\ & =\frac{1}{9}\int (27e^{3x}+9e^{-x}(-1+x)+\frac{32{(-1+e^3)}^8{x}^7}{e^{24}})dx\\ & =\frac{4{(1-e^3)}^8{x}^8}{9e^{24}}+3\int e^{3x}dx+\int e^{-x}(-1+x)dx\\ & =e^{3x}+e^{-x}(1-x)+\frac{4{(1-e^3)}^8{x}^8}{9e^{24}}+\int e^{-x}dx\\ & =-e^{-x}+e^{3x}+e^{-x}(1-x)+\frac{4{(1-e^3)}^8{x}^8}{9e^{24}}\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 31, normalized size = 1.00 $$\begin{array}{c}{e}^{3x}-e^{-x}x+\frac{4{(-1+e^3)}^8{x}^8}{9e^{24}}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(-24 - x)*(27*E^(24 + 4*x) + E^24*(-9 + 9*x) + E^x*(32*x^7 - 256*E^3*x^7 + 896*E^6*x^7 - 1792*E^9
*x^7 + 2240*E^12*x^7 - 1792*E^15*x^7 + 896*E^18*x^7 - 256*E^21*x^7 + 32*E^24*x^7)))/9,x]

[Out]

E^(3*x) - x/E^x + (4*(-1 + E^3)^8*x^8)/(9*E^24)

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fricas [B]  time = 1.37, size = 85, normalized size = 2.74 $$\begin{array}{c}-\frac{1}{9}(9x{e}^{24}-4(x^8{e}^{24}-8x^8{e}^{21}+28x^8{e}^{18}-56x^8{e}^{15}+70x^8{e}^{12}-56x^8{e}^9+28x^8{e}^6-8x^8{e}^3+x^8)e^x-9e^{(4x+24)})e^{(-x-24)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="fricas")

[Out]

-1/9*(9*x*e^24 - 4*(x^8*e^24 - 8*x^8*e^21 + 28*x^8*e^18 - 56*x^8*e^15 + 70*x^8*e^12 - 56*x^8*e^9 + 28*x^8*e^6
- 8*x^8*e^3 + x^8)*e^x - 9*e^(4*x + 24))*e^(-x - 24)

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giac [B]  time = 0.16, size = 85, normalized size = 2.74 $$\begin{array}{c}\frac{1}{9}(4x^8{e}^{108}-32x^8{e}^{105}+112x^8{e}^{102}-224x^8{e}^{99}+280x^8{e}^{96}-224x^8{e}^{93}+112x^8{e}^{90}-32x^8{e}^{87}+4x^8{e}^{84}-9x{e}^{(-x+108)}+9e^{(3x+108)})e^{(-108)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="giac")

[Out]

1/9*(4*x^8*e^108 - 32*x^8*e^105 + 112*x^8*e^102 - 224*x^8*e^99 + 280*x^8*e^96 - 224*x^8*e^93 + 112*x^8*e^90 -
32*x^8*e^87 + 4*x^8*e^84 - 9*x*e^(-x + 108) + 9*e^(3*x + 108))*e^(-108)

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maple [B]  time = 0.03, size = 92, normalized size = 2.97

method	result	size
risch	$-\frac{32{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^3}{9}+\frac{4{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^{24}}{9}-\frac{32{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^{21}}{9}+\frac{112{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^{18}}{9}-\frac{224{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^{15}}{9}+\frac{280{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^{12}}{9}-\frac{224{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^9}{9}+\frac{112{\mathrm{e}}^{-24}{x}^8{\mathrm{e}}^6}{9}+\frac{4{\mathrm{e}}^{-24}{x}^8}{9}+{\mathrm{e}}^{3x}-x{\mathrm{e}}^{-x}$	$92$
default	$\frac{{\mathrm{e}}^{-24}(4x^8-32x^8{\mathrm{e}}^3+112x^8{\mathrm{e}}^6-224x^8{\mathrm{e}}^9+280x^8{\mathrm{e}}^{12}-224x^8{\mathrm{e}}^{15}+112x^8{\mathrm{e}}^{18}-32x^8{\mathrm{e}}^{21}+4x^8{\mathrm{e}}^{24}+9{\mathrm{e}}^{-x}{\mathrm{e}}^{24}+9{\mathrm{e}}^{3x}{\mathrm{e}}^{24}+9{\mathrm{e}}^{24}(-x{\mathrm{e}}^{-x}-{\mathrm{e}}^{-x}))}{9}$	$123$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*x^7*ex
p(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x),x,me
thod=_RETURNVERBOSE)

[Out]

-32/9*exp(-24)*x^8*exp(3)+4/9*exp(-24)*x^8*exp(24)-32/9*exp(-24)*x^8*exp(21)+112/9*exp(-24)*x^8*exp(18)-224/9*
exp(-24)*x^8*exp(15)+280/9*exp(-24)*x^8*exp(12)-224/9*exp(-24)*x^8*exp(9)+112/9*exp(-24)*x^8*exp(6)+4/9*exp(-2
4)*x^8+exp(3*x)-x*exp(-x)

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maxima [B]  time = 0.36, size = 79, normalized size = 2.55 $$\begin{array}{c}-\frac{32}{9}{x}^8{e}^{(-3)}+\frac{112}{9}{x}^8{e}^{(-6)}-\frac{224}{9}{x}^8{e}^{(-9)}+\frac{280}{9}{x}^8{e}^{(-12)}-\frac{224}{9}{x}^8{e}^{(-15)}+\frac{112}{9}{x}^8{e}^{(-18)}-\frac{32}{9}{x}^8{e}^{(-21)}+\frac{4}{9}{x}^8{e}^{(-24)}+\frac{4}{9}{x}^8-(x+1)e^{(-x)}+e^{\left(3x\right)}+e^{(-x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)^8*exp(4*x)+(32*x^7*exp(3)^8-256*x^7*exp(3)^7+896*x^7*exp(3)^6-1792*x^7*exp(3)^5+2240*
x^7*exp(3)^4-1792*x^7*exp(3)^3+896*x^7*exp(3)^2-256*x^7*exp(3)+32*x^7)*exp(x)+(9*x-9)*exp(3)^8)/exp(3)^8/exp(x
),x, algorithm="maxima")

[Out]

-32/9*x^8*e^(-3) + 112/9*x^8*e^(-6) - 224/9*x^8*e^(-9) + 280/9*x^8*e^(-12) - 224/9*x^8*e^(-15) + 112/9*x^8*e^(
-18) - 32/9*x^8*e^(-21) + 4/9*x^8*e^(-24) + 4/9*x^8 - (x + 1)*e^(-x) + e^(3*x) + e^(-x)

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mupad [B]  time = 3.15, size = 50, normalized size = 1.61 $$\begin{array}{c}{\mathrm{e}}^{3x}-x{\mathrm{e}}^{-x}+x^8(\frac{112{\mathrm{e}}^{-6}}{9}-\frac{32{\mathrm{e}}^{-3}}{9}-\frac{224{\mathrm{e}}^{-9}}{9}+\frac{280{\mathrm{e}}^{-12}}{9}-\frac{224{\mathrm{e}}^{-15}}{9}+\frac{112{\mathrm{e}}^{-18}}{9}-\frac{32{\mathrm{e}}^{-21}}{9}+\frac{4{\mathrm{e}}^{-24}}{9}+\frac{4}{9})\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*exp(-24)*(3*exp(4*x)*exp(24) + (exp(x)*(896*x^7*exp(6) - 256*x^7*exp(3) - 1792*x^7*exp(9) + 2240*x
^7*exp(12) - 1792*x^7*exp(15) + 896*x^7*exp(18) - 256*x^7*exp(21) + 32*x^7*exp(24) + 32*x^7))/9 + (exp(24)*(9*
x - 9))/9),x)

[Out]

exp(3*x) - x*exp(-x) + x^8*((112*exp(-6))/9 - (32*exp(-3))/9 - (224*exp(-9))/9 + (280*exp(-12))/9 - (224*exp(-
15))/9 + (112*exp(-18))/9 - (32*exp(-21))/9 + (4*exp(-24))/9 + 4/9)

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sympy [B]  time = 0.23, size = 60, normalized size = 1.94 $$\begin{array}{c}\frac{x^8(-32e^{21}-224e^{15}-224e^9-32e^3+4+112e^6+280e^{12}+112e^{18}+4e^{24})}{9e^{24}}-x{e}^{-x}+e^{3x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(27*exp(3)**8*exp(4*x)+(32*x**7*exp(3)**8-256*x**7*exp(3)**7+896*x**7*exp(3)**6-1792*x**7*exp(3)
**5+2240*x**7*exp(3)**4-1792*x**7*exp(3)**3+896*x**7*exp(3)**2-256*x**7*exp(3)+32*x**7)*exp(x)+(9*x-9)*exp(3)*
*8)/exp(3)**8/exp(x),x)

[Out]

x**8*(-32*exp(21) - 224*exp(15) - 224*exp(9) - 32*exp(3) + 4 + 112*exp(6) + 280*exp(12) + 112*exp(18) + 4*exp(
24))*exp(-24)/9 - x*exp(-x) + exp(3*x)

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