3.41.37 $\int \frac{1}{9}(-9+e^{8x}(-250-2000x)+e^{12x}(-96-1152x)+e^{6x}(-48-288x)+e^{16x}(-1-16x)+e^{14x}(16+224x)+e^{4x}(96+384x)+e^{10x}(256+2560x))dx$

Optimal. Leaf size=29 $$3-\frac{1}{9}{(3-e^{4x}{(4-e^{2x})}^2)}^{2}x$$

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Rubi [B]  time = 0.10, antiderivative size = 165, normalized size of antiderivative = 5.69, number of steps used = 16, number of rules used = 3, integrand size = 83, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.036, Rules used = {12, 2176, 2194} $$\begin{array}{c}-x-\frac{8e^{4x}}{3}+\frac{8e^{6x}}{9}+\frac{125e^{8x}}{36}-\frac{128e^{10x}}{45}+\frac{8e^{12x}}{9}-\frac{8e^{14x}}{63}+\frac{e^{16x}}{144}+\frac{8}{3}{e}^{4x}(4x+1)-\frac{8}{9}{e}^{6x}(6x+1)-\frac{125}{36}{e}^{8x}(8x+1)+\frac{128}{45}{e}^{10x}(10x+1)-\frac{8}{9}{e}^{12x}(12x+1)+\frac{8}{63}{e}^{14x}(14x+1)-\frac{1}{144}{e}^{16x}(16x+1)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-9 + E^(8*x)*(-250 - 2000*x) + E^(12*x)*(-96 - 1152*x) + E^(6*x)*(-48 - 288*x) + E^(16*x)*(-1 - 16*x) + E
^(14*x)*(16 + 224*x) + E^(4*x)*(96 + 384*x) + E^(10*x)*(256 + 2560*x))/9,x]

[Out]

(-8*E^(4*x))/3 + (8*E^(6*x))/9 + (125*E^(8*x))/36 - (128*E^(10*x))/45 + (8*E^(12*x))/9 - (8*E^(14*x))/63 + E^(
16*x)/144 - x + (8*E^(4*x)*(1 + 4*x))/3 - (8*E^(6*x)*(1 + 6*x))/9 - (125*E^(8*x)*(1 + 8*x))/36 + (128*E^(10*x)
*(1 + 10*x))/45 - (8*E^(12*x)*(1 + 12*x))/9 + (8*E^(14*x)*(1 + 14*x))/63 - (E^(16*x)*(1 + 16*x))/144

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]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{1}{9}\int (-9+e^{8x}(-250-2000x)+e^{12x}(-96-1152x)+e^{6x}(-48-288x)+e^{16x}(-1-16x)+e^{14x}(16+224x)+e^{4x}(96+384x)+e^{10x}(256+2560x))dx\\ & =-x+\frac{1}{9}\int e^{8x}(-250-2000x)dx+\frac{1}{9}\int e^{12x}(-96-1152x)dx+\frac{1}{9}\int e^{6x}(-48-288x)dx+\frac{1}{9}\int e^{16x}(-1-16x)dx+\frac{1}{9}\int e^{14x}(16+224x)dx+\frac{1}{9}\int e^{4x}(96+384x)dx+\frac{1}{9}\int e^{10x}(256+2560x)dx\\ & =-x+\frac{8}{3}{e}^{4x}(1+4x)-\frac{8}{9}{e}^{6x}(1+6x)-\frac{125}{36}{e}^{8x}(1+8x)+\frac{128}{45}{e}^{10x}(1+10x)-\frac{8}{9}{e}^{12x}(1+12x)+\frac{8}{63}{e}^{14x}(1+14x)-\frac{1}{144}{e}^{16x}(1+16x)+\frac{1}{9}\int e^{16x}dx-\frac{16}{9}\int e^{14x}dx+\frac{16}{3}\int e^{6x}dx-\frac{32}{3}\int e^{4x}dx+\frac{32}{3}\int e^{12x}dx+\frac{250}{9}\int e^{8x}dx-\frac{256}{9}\int e^{10x}dx\\ & =-\frac{8e^{4x}}{3}+\frac{8e^{6x}}{9}+\frac{125e^{8x}}{36}-\frac{128e^{10x}}{45}+\frac{8e^{12x}}{9}-\frac{8e^{14x}}{63}+\frac{e^{16x}}{144}-x+\frac{8}{3}{e}^{4x}(1+4x)-\frac{8}{9}{e}^{6x}(1+6x)-\frac{125}{36}{e}^{8x}(1+8x)+\frac{128}{45}{e}^{10x}(1+10x)-\frac{8}{9}{e}^{12x}(1+12x)+\frac{8}{63}{e}^{14x}(1+14x)-\frac{1}{144}{e}^{16x}(1+16x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.12, size = 28, normalized size = 0.97 $$\begin{array}{c}-\frac{1}{9}{(-3+16e^{4x}-8e^{6x}+e^{8x})}^{2}x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-9 + E^(8*x)*(-250 - 2000*x) + E^(12*x)*(-96 - 1152*x) + E^(6*x)*(-48 - 288*x) + E^(16*x)*(-1 - 16*
x) + E^(14*x)*(16 + 224*x) + E^(4*x)*(96 + 384*x) + E^(10*x)*(256 + 2560*x))/9,x]

[Out]

-1/9*((-3 + 16*E^(4*x) - 8*E^(6*x) + E^(8*x))^2*x)

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fricas [B]  time = 0.54, size = 53, normalized size = 1.83 $$\begin{array}{c}-\frac{1}{9}x{e}^{\left(16x\right)}+\frac{16}{9}x{e}^{\left(14x\right)}-\frac{32}{3}x{e}^{\left(12x\right)}+\frac{256}{9}x{e}^{\left(10x\right)}-\frac{250}{9}x{e}^{\left(8x\right)}-\frac{16}{3}x{e}^{\left(6x\right)}+\frac{32}{3}x{e}^{\left(4x\right)}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-16*x-1)*exp(x)^16+1/9*(224*x+16)*exp(x)^14+1/9*(-1152*x-96)*exp(x)^12+1/9*(2560*x+256)*exp(x)^
10+1/9*(-2000*x-250)*exp(x)^8+1/9*(-288*x-48)*exp(x)^6+1/9*(384*x+96)*exp(x)^4-1,x, algorithm="fricas")

[Out]

-1/9*x*e^(16*x) + 16/9*x*e^(14*x) - 32/3*x*e^(12*x) + 256/9*x*e^(10*x) - 250/9*x*e^(8*x) - 16/3*x*e^(6*x) + 32
/3*x*e^(4*x) - x

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giac [B]  time = 0.14, size = 53, normalized size = 1.83 $$\begin{array}{c}-\frac{1}{9}x{e}^{\left(16x\right)}+\frac{16}{9}x{e}^{\left(14x\right)}-\frac{32}{3}x{e}^{\left(12x\right)}+\frac{256}{9}x{e}^{\left(10x\right)}-\frac{250}{9}x{e}^{\left(8x\right)}-\frac{16}{3}x{e}^{\left(6x\right)}+\frac{32}{3}x{e}^{\left(4x\right)}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-16*x-1)*exp(x)^16+1/9*(224*x+16)*exp(x)^14+1/9*(-1152*x-96)*exp(x)^12+1/9*(2560*x+256)*exp(x)^
10+1/9*(-2000*x-250)*exp(x)^8+1/9*(-288*x-48)*exp(x)^6+1/9*(384*x+96)*exp(x)^4-1,x, algorithm="giac")

[Out]

-1/9*x*e^(16*x) + 16/9*x*e^(14*x) - 32/3*x*e^(12*x) + 256/9*x*e^(10*x) - 250/9*x*e^(8*x) - 16/3*x*e^(6*x) + 32
/3*x*e^(4*x) - x

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maple [B]  time = 0.04, size = 54, normalized size = 1.86

method	result	size
default	$-x-\frac{250x{\mathrm{e}}^{8x}}{9}-\frac{32{\mathrm{e}}^{12x}x}{3}-\frac{16x{\mathrm{e}}^{6x}}{3}-\frac{{\mathrm{e}}^{16x}x}{9}+\frac{16{\mathrm{e}}^{14x}x}{9}+\frac{32x{\mathrm{e}}^{4x}}{3}+\frac{256{\mathrm{e}}^{10x}x}{9}$	$54$
risch	$-x-\frac{250x{\mathrm{e}}^{8x}}{9}-\frac{32{\mathrm{e}}^{12x}x}{3}-\frac{16x{\mathrm{e}}^{6x}}{3}-\frac{{\mathrm{e}}^{16x}x}{9}+\frac{16{\mathrm{e}}^{14x}x}{9}+\frac{32x{\mathrm{e}}^{4x}}{3}+\frac{256{\mathrm{e}}^{10x}x}{9}$	$54$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*(-16*x-1)*exp(x)^16+1/9*(224*x+16)*exp(x)^14+1/9*(-1152*x-96)*exp(x)^12+1/9*(2560*x+256)*exp(x)^10+1/9
*(-2000*x-250)*exp(x)^8+1/9*(-288*x-48)*exp(x)^6+1/9*(384*x+96)*exp(x)^4-1,x,method=_RETURNVERBOSE)

[Out]

-x-250/9*x*exp(x)^8-32/3*exp(x)^12*x-16/3*x*exp(x)^6-1/9*exp(x)^16*x+16/9*exp(x)^14*x+32/3*x*exp(x)^4+256/9*ex
p(x)^10*x

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maxima [B]  time = 0.37, size = 63, normalized size = 2.17 $$\begin{array}{c}-\frac{1}{9}x{e}^{\left(16x\right)}+\frac{16}{9}x{e}^{\left(14x\right)}-\frac{32}{3}x{e}^{\left(12x\right)}+\frac{256}{9}x{e}^{\left(10x\right)}-\frac{250}{9}x{e}^{\left(8x\right)}-\frac{16}{3}x{e}^{\left(6x\right)}+\frac{8}{3}(4x-1)e^{\left(4x\right)}-x+\frac{8}{3}{e}^{\left(4x\right)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-16*x-1)*exp(x)^16+1/9*(224*x+16)*exp(x)^14+1/9*(-1152*x-96)*exp(x)^12+1/9*(2560*x+256)*exp(x)^
10+1/9*(-2000*x-250)*exp(x)^8+1/9*(-288*x-48)*exp(x)^6+1/9*(384*x+96)*exp(x)^4-1,x, algorithm="maxima")

[Out]

-1/9*x*e^(16*x) + 16/9*x*e^(14*x) - 32/3*x*e^(12*x) + 256/9*x*e^(10*x) - 250/9*x*e^(8*x) - 16/3*x*e^(6*x) + 8/
3*(4*x - 1)*e^(4*x) - x + 8/3*e^(4*x)

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mupad [B]  time = 3.09, size = 23, normalized size = 0.79 $$\begin{array}{c}-\frac{x{(16{\mathrm{e}}^{4x}-8{\mathrm{e}}^{6x}+{\mathrm{e}}^{8x}-3)}^2}{9}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(14*x)*(224*x + 16))/9 - (exp(16*x)*(16*x + 1))/9 - (exp(6*x)*(288*x + 48))/9 + (exp(4*x)*(384*x + 96)
)/9 - (exp(12*x)*(1152*x + 96))/9 - (exp(8*x)*(2000*x + 250))/9 + (exp(10*x)*(2560*x + 256))/9 - 1,x)

[Out]

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

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sympy [B]  time = 0.21, size = 70, normalized size = 2.41 $$\begin{array}{c}-\frac{x{e}^{16x}}{9}+\frac{16x{e}^{14x}}{9}-\frac{32x{e}^{12x}}{3}+\frac{256x{e}^{10x}}{9}-\frac{250x{e}^{8x}}{9}-\frac{16x{e}^{6x}}{3}+\frac{32x{e}^{4x}}{3}-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*(-16*x-1)*exp(x)**16+1/9*(224*x+16)*exp(x)**14+1/9*(-1152*x-96)*exp(x)**12+1/9*(2560*x+256)*exp(
x)**10+1/9*(-2000*x-250)*exp(x)**8+1/9*(-288*x-48)*exp(x)**6+1/9*(384*x+96)*exp(x)**4-1,x)

[Out]

-x*exp(16*x)/9 + 16*x*exp(14*x)/9 - 32*x*exp(12*x)/3 + 256*x*exp(10*x)/9 - 250*x*exp(8*x)/9 - 16*x*exp(6*x)/3
+ 32*x*exp(4*x)/3 - x

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