3.30.12 \(\int \frac {e^{\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}} (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} (-4-3 x^2+4 x^3+3 x^4+e^8 (-8 x+8 x^2+6 x^3-4 x^4-2 x^5)))}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {\left (-2+x+x^2\right )^2}{\left (1-e^{e^8 x}\right )^2 x}} \]

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Rubi [F]  time = 25.62, 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 {\exp \left (\frac {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}\right ) \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4 - 4*x - 3*x^2 + 2*x^3 + x^4)/(x - 2*E^(E^8*x)*x + E^(2*E^8*x)*x))*(4 + 3*x^2 - 4*x^3 - 3*x^4 + E^(E
^8*x)*(-4 - 3*x^2 + 4*x^3 + 3*x^4 + E^8*(-8*x + 8*x^2 + 6*x^3 - 4*x^4 - 2*x^5))))/(-x^2 + 3*E^(E^8*x)*x^2 - 3*
E^(2*E^8*x)*x^2 + E^(3*E^8*x)*x^2),x]

[Out]

8*Defer[Int][E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))/(-1 + E^(E^8*x))^3, x] - (3 - 8*E^8)*Defer[Int][E
^((-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))/(-1 + E^(E^8*x))^2, x] - 4*Defer[Int][E^((-2 + x + x^2)^2/((-1 + E^
(E^8*x))^2*x))/((-1 + E^(E^8*x))^2*x^2), x] - 8*Defer[Int][E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))/((-
1 + E^(E^8*x))^3*x), x] - 8*Defer[Int][E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))/((-1 + E^(E^8*x))^2*x),
x] + 6*Defer[Int][(E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))*x)/(-1 + E^(E^8*x))^3, x] + 2*(2 + 3*E^8)*
Defer[Int][(E^((-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))*x)/(-1 + E^(E^8*x))^2, x] - 4*Defer[Int][(E^(8 + (-2 +
x + x^2)^2/((-1 + E^(E^8*x))^2*x))*x^2)/(-1 + E^(E^8*x))^3, x] + (3 - 4*E^8)*Defer[Int][(E^((-2 + x + x^2)^2/
((-1 + E^(E^8*x))^2*x))*x^2)/(-1 + E^(E^8*x))^2, x] - 2*Defer[Int][(E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^
2*x))*x^3)/(-1 + E^(E^8*x))^3, x] - 2*Defer[Int][(E^(8 + (-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))*x^3)/(-1 + E
^(E^8*x))^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (2-x-x^2\right ) \left (-2-x-3 x^2-2 e^{8+e^8 x} x \left (-2+x+x^2\right )+e^{e^8 x} \left (2+x+3 x^2\right )\right )}{\left (1-e^{e^8 x}\right )^3 x^2} \, dx\\ &=\int \left (-\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^3 x}+\frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-8 e^8 x-3 \left (1-\frac {8 e^8}{3}\right ) x^2+4 \left (1+\frac {3 e^8}{2}\right ) x^3+3 \left (1-\frac {4 e^8}{3}\right ) x^4-2 e^8 x^5\right )}{\left (1-e^{e^8 x}\right )^2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^3 x} \, dx\right )+\int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-8 e^8 x-3 \left (1-\frac {8 e^8}{3}\right ) x^2+4 \left (1+\frac {3 e^8}{2}\right ) x^3+3 \left (1-\frac {4 e^8}{3}\right ) x^4-2 e^8 x^5\right )}{\left (1-e^{e^8 x}\right )^2 x^2} \, dx\\ &=-\left (2 \int \left (-\frac {4 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3}+\frac {4 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x}-\frac {3 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3}+\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3}+\frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3}\right ) \, dx\right )+\int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-3 x^2+4 x^3+3 x^4-2 e^8 x \left (-2+x+x^2\right )^2\right )}{\left (1-e^{e^8 x}\right )^2 x^2} \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3} \, dx\right )-4 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3} \, dx+6 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3} \, dx+8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x} \, dx+\int \left (-\frac {3 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (1-\frac {8 e^8}{3}\right )}{\left (-1+e^{e^8 x}\right )^2}-\frac {4 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x^2}-\frac {8 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x}+\frac {2 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (2+3 e^8\right ) x}{\left (-1+e^{e^8 x}\right )^2}-\frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-3+4 e^8\right ) x^2}{\left (-1+e^{e^8 x}\right )^2}-\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3} \, dx\right )-2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^2} \, dx-4 \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x^2} \, dx-4 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3} \, dx+6 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3} \, dx+8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x} \, dx-\left (3-8 e^8\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2} \, dx+\left (3-4 e^8\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^2} \, dx+\left (2 \left (2+3 e^8\right )\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 25, normalized size = 0.93 \begin {gather*} e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((4 - 4*x - 3*x^2 + 2*x^3 + x^4)/(x - 2*E^(E^8*x)*x + E^(2*E^8*x)*x))*(4 + 3*x^2 - 4*x^3 - 3*x^4
+ E^(E^8*x)*(-4 - 3*x^2 + 4*x^3 + 3*x^4 + E^8*(-8*x + 8*x^2 + 6*x^3 - 4*x^4 - 2*x^5))))/(-x^2 + 3*E^(E^8*x)*x^
2 - 3*E^(2*E^8*x)*x^2 + E^(3*E^8*x)*x^2),x]

[Out]

E^((-2 + x + x^2)^2/((-1 + E^(E^8*x))^2*x))

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fricas [A]  time = 0.80, size = 40, normalized size = 1.48 \begin {gather*} e^{\left (\frac {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x + 4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*exp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*
exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*exp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*ex
p(4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 7.36, size = 121, normalized size = 4.48 \begin {gather*} e^{\left (\frac {x^{4}}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} + \frac {2 \, x^{3}}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} - \frac {3 \, x^{2}}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} - \frac {4 \, x}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} + \frac {4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*exp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*
exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*exp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*ex
p(4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.22, size = 33, normalized size = 1.22

method	result	size
risch	\({\mathrm e}^{\frac {\left (2+x \right )^{2} \left (x -1\right )^{2}}{x \left ({\mathrm e}^{2 x \,{\mathrm e}^{8}}-2 \,{\mathrm e}^{x \,{\mathrm e}^{8}}+1\right )}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*exp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*exp((x
^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*exp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*exp(4)^2
)^2+3*x^2*exp(x*exp(4)^2)-x^2),x,method=_RETURNVERBOSE)

[Out]

exp((2+x)^2*(x-1)^2/x/(exp(2*x*exp(8))-2*exp(x*exp(8))+1))

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maxima [B]  time = 0.91, size = 106, normalized size = 3.93 \begin {gather*} e^{\left (\frac {x^{3}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {2 \, x^{2}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} - \frac {3 \, x}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} - \frac {4}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^5-4*x^4+6*x^3+8*x^2-8*x)*exp(4)^2+3*x^4+4*x^3-3*x^2-4)*exp(x*exp(4)^2)-3*x^4-4*x^3+3*x^2+4)*
exp((x^4+2*x^3-3*x^2-4*x+4)/(x*exp(x*exp(4)^2)^2-2*x*exp(x*exp(4)^2)+x))/(x^2*exp(x*exp(4)^2)^3-3*x^2*exp(x*ex
p(4)^2)^2+3*x^2*exp(x*exp(4)^2)-x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.22, size = 110, normalized size = 4.07 \begin {gather*} {\mathrm {e}}^{-\frac {4}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{-\frac {3\,x}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {4}{x-2\,x\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}}}\,{\mathrm {e}}^{\frac {x^3}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {2\,x^2}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x^3 - 3*x^2 - 4*x + x^4 + 4)/(x - 2*x*exp(x*exp(8)) + x*exp(2*x*exp(8))))*(4*x^3 - 3*x^2 + 3*x^4 +
exp(x*exp(8))*(exp(8)*(8*x - 8*x^2 - 6*x^3 + 4*x^4 + 2*x^5) + 3*x^2 - 4*x^3 - 3*x^4 + 4) - 4))/(x^2 - 3*x^2*e
xp(x*exp(8)) + 3*x^2*exp(2*x*exp(8)) - x^2*exp(3*x*exp(8))),x)

[Out]

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

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sympy [B]  time = 0.86, size = 41, normalized size = 1.52 \begin {gather*} e^{\frac {x^{4} + 2 x^{3} - 3 x^{2} - 4 x + 4}{x e^{2 x e^{8}} - 2 x e^{x e^{8}} + x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**5-4*x**4+6*x**3+8*x**2-8*x)*exp(4)**2+3*x**4+4*x**3-3*x**2-4)*exp(x*exp(4)**2)-3*x**4-4*x**
3+3*x**2+4)*exp((x**4+2*x**3-3*x**2-4*x+4)/(x*exp(x*exp(4)**2)**2-2*x*exp(x*exp(4)**2)+x))/(x**2*exp(x*exp(4)*
*2)**3-3*x**2*exp(x*exp(4)**2)**2+3*x**2*exp(x*exp(4)**2)-x**2),x)

[Out]

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

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