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

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

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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 {gather*} -x-\frac {8 e^{4 x}}{3}+\frac {8 e^{6 x}}{9}+\frac {125 e^{8 x}}{36}-\frac {128 e^{10 x}}{45}+\frac {8 e^{12 x}}{9}-\frac {8 e^{14 x}}{63}+\frac {e^{16 x}}{144}+\frac {8}{3} e^{4 x} (4 x+1)-\frac {8}{9} e^{6 x} (6 x+1)-\frac {125}{36} e^{8 x} (8 x+1)+\frac {128}{45} e^{10 x} (10 x+1)-\frac {8}{9} e^{12 x} (12 x+1)+\frac {8}{63} e^{14 x} (14 x+1)-\frac {1}{144} e^{16 x} (16 x+1) \end {gather*}

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

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

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 {gather*} -\frac {1}{9} \, x e^{\left (16 \, x\right )} + \frac {16}{9} \, x e^{\left (14 \, x\right )} - \frac {32}{3} \, x e^{\left (12 \, x\right )} + \frac {256}{9} \, x e^{\left (10 \, x\right )} - \frac {250}{9} \, x e^{\left (8 \, x\right )} - \frac {16}{3} \, x e^{\left (6 \, x\right )} + \frac {32}{3} \, x e^{\left (4 \, x\right )} - x \end {gather*}

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 {gather*} -\frac {1}{9} \, x e^{\left (16 \, x\right )} + \frac {16}{9} \, x e^{\left (14 \, x\right )} - \frac {32}{3} \, x e^{\left (12 \, x\right )} + \frac {256}{9} \, x e^{\left (10 \, x\right )} - \frac {250}{9} \, x e^{\left (8 \, x\right )} - \frac {16}{3} \, x e^{\left (6 \, x\right )} + \frac {32}{3} \, x e^{\left (4 \, x\right )} - x \end {gather*}

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	\(-x -\frac {250 x \,{\mathrm e}^{8 x}}{9}-\frac {32 \,{\mathrm e}^{12 x} x}{3}-\frac {16 x \,{\mathrm e}^{6 x}}{3}-\frac {{\mathrm e}^{16 x} x}{9}+\frac {16 \,{\mathrm e}^{14 x} x}{9}+\frac {32 x \,{\mathrm e}^{4 x}}{3}+\frac {256 \,{\mathrm e}^{10 x} x}{9}\)	\(54\)
risch	\(-x -\frac {250 x \,{\mathrm e}^{8 x}}{9}-\frac {32 \,{\mathrm e}^{12 x} x}{3}-\frac {16 x \,{\mathrm e}^{6 x}}{3}-\frac {{\mathrm e}^{16 x} x}{9}+\frac {16 \,{\mathrm e}^{14 x} x}{9}+\frac {32 x \,{\mathrm e}^{4 x}}{3}+\frac {256 \,{\mathrm e}^{10 x} x}{9}\)	\(54\)

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 {gather*} -\frac {1}{9} \, x e^{\left (16 \, x\right )} + \frac {16}{9} \, x e^{\left (14 \, x\right )} - \frac {32}{3} \, x e^{\left (12 \, x\right )} + \frac {256}{9} \, x e^{\left (10 \, x\right )} - \frac {250}{9} \, x e^{\left (8 \, x\right )} - \frac {16}{3} \, x e^{\left (6 \, x\right )} + \frac {8}{3} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} - x + \frac {8}{3} \, e^{\left (4 \, x\right )} \end {gather*}

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 {gather*} -\frac {x\,{\left (16\,{\mathrm {e}}^{4\,x}-8\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}-3\right )}^2}{9} \end {gather*}

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 {gather*} - \frac {x e^{16 x}}{9} + \frac {16 x e^{14 x}}{9} - \frac {32 x e^{12 x}}{3} + \frac {256 x e^{10 x}}{9} - \frac {250 x e^{8 x}}{9} - \frac {16 x e^{6 x}}{3} + \frac {32 x e^{4 x}}{3} - x \end {gather*}

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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