∫ e−e1+9x+x (−3+27e1+9x)+24x__ e−2e1+9x+2x−8e−e1+9x+xx2+16x4 dx

3.39.22 \(\int \frac {e^{-e^{1+9 x}+x} (-3+27 e^{1+9 x})+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx\)

Optimal. Leaf size=23 \[ \frac {3}{e^{-e^{1+9 x}+x}-4 x^2} \]

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Rubi [F] time = 4.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x}{e^{-2 e^{1+9 x}+2 x}-8 e^{-e^{1+9 x}+x} x^2+16 x^4} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(E^(-E^(1 + 9*x) + x)*(-3 + 27*E^(1 + 9*x)) + 24*x)/(E^(-2*E^(1 + 9*x) + 2*x) - 8*E^(-E^(1 + 9*x) + x)*x^2

+ 16*x^4),x]

[Out]

(-3*E^(1/9 + 8*x)*Gamma[8/9, -E^(1 + 9*x)])/(-E^(9*x))^(8/9) + 216*Defer[Int][E^(1 + 2*E^(1 + 9*x) + 7*x)*x^2,

x] + 1296*Defer[Int][E^(1 + 3*E^(1 + 9*x) + 6*x)*x^4, x] + 6912*Defer[Int][E^(1 + 4*E^(1 + 9*x) + 5*x)*x^6, x

] + 34560*Defer[Int][E^(1 + 5*E^(1 + 9*x) + 4*x)*x^8, x] + 165888*Defer[Int][E^(1 + 6*E^(1 + 9*x) + 3*x)*x^10,

x] + 774144*Defer[Int][E^(1 + 7*E^(1 + 9*x) + 2*x)*x^12, x] + 3538944*Defer[Int][E^(1 + 8*E^(1 + 9*x) + x)*x^

14, x] + 15925248*Defer[Int][E^(1 + 9*E^(1 + 9*x))*x^16, x] - 3*Defer[Int][E^E^(1 + 9*x)/(E^x - 4*E^E^(1 + 9*x

)*x^2), x] + 24*Defer[Int][(E^(2*E^(1 + 9*x))*x)/(-E^x + 4*E^E^(1 + 9*x)*x^2)^2, x] - 12*Defer[Int][(E^(2*E^(1

+ 9*x))*x^2)/(-E^x + 4*E^E^(1 + 9*x)*x^2)^2, x] + 28311552*Defer[Int][(E^(1 + 11*E^(1 + 9*x))*x^20)/(-E^x + 4

*E^E^(1 + 9*x)*x^2)^2, x] - 70778880*Defer[Int][(E^(1 + 10*E^(1 + 9*x))*x^18)/(-E^x + 4*E^E^(1 + 9*x)*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 e^{1+9 x}} \left (e^{-e^{1+9 x}+x} \left (-3+27 e^{1+9 x}\right )+24 x\right )}{\left (e^x-4 e^{e^{1+9 x}} x^2\right )^2} \, dx\\ &=\int \left (27 e^{1+e^{1+9 x}+8 x}+216 e^{1+2 e^{1+9 x}+7 x} x^2+1296 e^{1+3 e^{1+9 x}+6 x} x^4+6912 e^{1+4 e^{1+9 x}+5 x} x^6+34560 e^{1+5 e^{1+9 x}+4 x} x^8+165888 e^{1+6 e^{1+9 x}+3 x} x^{10}+774144 e^{1+7 e^{1+9 x}+2 x} x^{12}+3538944 e^{1+8 e^{1+9 x}+x} x^{14}+15925248 e^{1+9 e^{1+9 x}} x^{16}-\frac {3 e^{e^{1+9 x}} \left (-1+23592960 e^{1+9 e^{1+9 x}} x^{18}\right )}{-e^x+4 e^{e^{1+9 x}} x^2}+\frac {12 e^{2 e^{1+9 x}} x \left (2-x+2359296 e^{1+9 e^{1+9 x}} x^{19}\right )}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {e^{e^{1+9 x}} \left (-1+23592960 e^{1+9 e^{1+9 x}} x^{18}\right )}{-e^x+4 e^{e^{1+9 x}} x^2} \, dx\right )+12 \int \frac {e^{2 e^{1+9 x}} x \left (2-x+2359296 e^{1+9 e^{1+9 x}} x^{19}\right )}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2} \, dx+27 \int e^{1+e^{1+9 x}+8 x} \, dx+216 \int e^{1+2 e^{1+9 x}+7 x} x^2 \, dx+1296 \int e^{1+3 e^{1+9 x}+6 x} x^4 \, dx+6912 \int e^{1+4 e^{1+9 x}+5 x} x^6 \, dx+34560 \int e^{1+5 e^{1+9 x}+4 x} x^8 \, dx+165888 \int e^{1+6 e^{1+9 x}+3 x} x^{10} \, dx+774144 \int e^{1+7 e^{1+9 x}+2 x} x^{12} \, dx+3538944 \int e^{1+8 e^{1+9 x}+x} x^{14} \, dx+15925248 \int e^{1+9 e^{1+9 x}} x^{16} \, dx\\ &=-\left (3 \int \left (\frac {e^{e^{1+9 x}}}{e^x-4 e^{e^{1+9 x}} x^2}+\frac {23592960 e^{1+10 e^{1+9 x}} x^{18}}{-e^x+4 e^{e^{1+9 x}} x^2}\right ) \, dx\right )+12 \int \left (\frac {2 e^{2 e^{1+9 x}} x}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2}-\frac {e^{2 e^{1+9 x}} x^2}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2}+\frac {2359296 e^{1+11 e^{1+9 x}} x^{20}}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2}\right ) \, dx+27 \operatorname {Subst}\left (\int e^{1+e x^9} x^7 \, dx,x,e^x\right )+216 \int e^{1+2 e^{1+9 x}+7 x} x^2 \, dx+1296 \int e^{1+3 e^{1+9 x}+6 x} x^4 \, dx+6912 \int e^{1+4 e^{1+9 x}+5 x} x^6 \, dx+34560 \int e^{1+5 e^{1+9 x}+4 x} x^8 \, dx+165888 \int e^{1+6 e^{1+9 x}+3 x} x^{10} \, dx+774144 \int e^{1+7 e^{1+9 x}+2 x} x^{12} \, dx+3538944 \int e^{1+8 e^{1+9 x}+x} x^{14} \, dx+15925248 \int e^{1+9 e^{1+9 x}} x^{16} \, dx\\ &=-\frac {3 e^{\frac {1}{9}+8 x} \Gamma \left (\frac {8}{9},-e^{1+9 x}\right )}{\left (-e^{9 x}\right )^{8/9}}-3 \int \frac {e^{e^{1+9 x}}}{e^x-4 e^{e^{1+9 x}} x^2} \, dx-12 \int \frac {e^{2 e^{1+9 x}} x^2}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2} \, dx+24 \int \frac {e^{2 e^{1+9 x}} x}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2} \, dx+216 \int e^{1+2 e^{1+9 x}+7 x} x^2 \, dx+1296 \int e^{1+3 e^{1+9 x}+6 x} x^4 \, dx+6912 \int e^{1+4 e^{1+9 x}+5 x} x^6 \, dx+34560 \int e^{1+5 e^{1+9 x}+4 x} x^8 \, dx+165888 \int e^{1+6 e^{1+9 x}+3 x} x^{10} \, dx+774144 \int e^{1+7 e^{1+9 x}+2 x} x^{12} \, dx+3538944 \int e^{1+8 e^{1+9 x}+x} x^{14} \, dx+15925248 \int e^{1+9 e^{1+9 x}} x^{16} \, dx+28311552 \int \frac {e^{1+11 e^{1+9 x}} x^{20}}{\left (-e^x+4 e^{e^{1+9 x}} x^2\right )^2} \, dx-70778880 \int \frac {e^{1+10 e^{1+9 x}} x^{18}}{-e^x+4 e^{e^{1+9 x}} x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.73, size = 31, normalized size = 1.35 \begin {gather*} \frac {3 e^{e^{1+9 x}}}{e^x-4 e^{e^{1+9 x}} x^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^(-E^(1 + 9*x) + x)*(-3 + 27*E^(1 + 9*x)) + 24*x)/(E^(-2*E^(1 + 9*x) + 2*x) - 8*E^(-E^(1 + 9*x) +

x)*x^2 + 16*x^4),x]

[Out]

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

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fricas [A] time = 1.31, size = 23, normalized size = 1.00 \begin {gather*} -\frac {3}{4 \, x^{2} - e^{\left (x - e^{\left (9 \, x + 1\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*e

xp(9*x)+x)+16*x^4),x, algorithm="fricas")

[Out]

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

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giac [B] time = 0.23, size = 242, normalized size = 10.52 \begin {gather*} -\frac {3 \, {\left (36 \, x^{3} e^{\left (9 \, x + \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - 4 \, x^{3} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 8 \, x^{2} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 9 \, x e^{\left (10 \, x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} + x e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 2 \, e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )}\right )}}{144 \, x^{5} e^{\left (9 \, x + \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - 16 \, x^{5} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 32 \, x^{4} e^{\left (\frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 72 \, x^{3} e^{\left (10 \, x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} + 8 \, x^{3} e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} - 16 \, x^{2} e^{\left (x - \frac {1}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 9 \, x e^{\left (11 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )} + 1\right )} - x e^{\left (2 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )}\right )} + 2 \, e^{\left (2 \, x - \frac {3}{2} \, e^{\left (9 \, x + 1\right )}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*e

xp(9*x)+x)+16*x^4),x, algorithm="giac")

[Out]

-3*(36*x^3*e^(9*x + 1/2*e^(9*x + 1) + 1) - 4*x^3*e^(1/2*e^(9*x + 1)) + 8*x^2*e^(1/2*e^(9*x + 1)) - 9*x*e^(10*x

- 1/2*e^(9*x + 1) + 1) + x*e^(x - 1/2*e^(9*x + 1)) - 2*e^(x - 1/2*e^(9*x + 1)))/(144*x^5*e^(9*x + 1/2*e^(9*x

+ 1) + 1) - 16*x^5*e^(1/2*e^(9*x + 1)) + 32*x^4*e^(1/2*e^(9*x + 1)) - 72*x^3*e^(10*x - 1/2*e^(9*x + 1) + 1) +

8*x^3*e^(x - 1/2*e^(9*x + 1)) - 16*x^2*e^(x - 1/2*e^(9*x + 1)) + 9*x*e^(11*x - 3/2*e^(9*x + 1) + 1) - x*e^(2*x

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

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maple [A] time = 0.15, size = 24, normalized size = 1.04


method	result	size

norman	\(-\frac {3}{4 x^{2}-{\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{9 x}+x}}\)	\(24\)
risch	\(-\frac {3}{4 x^{2}-{\mathrm e}^{-{\mathrm e}^{9 x +1}+x}}\)	\(24\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*exp(9*x

)+x)+16*x^4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A] time = 0.39, size = 28, normalized size = 1.22 \begin {gather*} -\frac {3 \, e^{\left (e^{\left (9 \, x + 1\right )}\right )}}{4 \, x^{2} e^{\left (e^{\left (9 \, x + 1\right )}\right )} - e^{x}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp(9*x)+x)^2-8*x^2*exp(-exp(1)*e

xp(9*x)+x)+16*x^4),x, algorithm="maxima")

[Out]

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

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mupad [B] time = 0.19, size = 21, normalized size = 0.91 \begin {gather*} \frac {3}{{\mathrm {e}}^{x-{\mathrm {e}}^{9\,x}\,\mathrm {e}}-4\,x^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((24*x + exp(x - exp(9*x)*exp(1))*(27*exp(9*x)*exp(1) - 3))/(exp(2*x - 2*exp(9*x)*exp(1)) - 8*x^2*exp(x - e

xp(9*x)*exp(1)) + 16*x^4),x)

[Out]

3/(exp(x - exp(9*x)*exp(1)) - 4*x^2)

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sympy [A] time = 0.15, size = 17, normalized size = 0.74 \begin {gather*} \frac {3}{- 4 x^{2} + e^{x - e e^{9 x}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*exp(1)*exp(9*x)-3)*exp(-exp(1)*exp(9*x)+x)+24*x)/(exp(-exp(1)*exp(9*x)+x)**2-8*x**2*exp(-exp(1)

*exp(9*x)+x)+16*x**4),x)

[Out]

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

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