\(\int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} (8 x^3-8 x^5+2 x^7+e^{-x} (4+2 x-4 x^2-x^3))}{4 x^3-4 x^5+x^7} \, dx\) [8692]

Optimal result

Integrand size = 77, antiderivative size = 26 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{2 x-\frac {e^{-x}}{2 x^2-x^4}} \]

[Out]

exp(2*x)/exp(exp(-x)/(-x^4+2*x^2))

Rubi [F]

\[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=\int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx \]

[In]

Int[(E^(2*x + 1/(E^x*(-2*x^2 + x^4)))*(8*x^3 - 8*x^5 + 2*x^7 + (4 + 2*x - 4*x^2 - x^3)/E^x))/(4*x^3 - 4*x^5 +
x^7),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{x^3 \left (4-4 x^2+x^4\right )} \, dx \\ & = \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{x^3 \left (-2+x^2\right )^2} \, dx \\ & = \int \left (2 e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} \left (-4-2 x+4 x^2+x^3\right )}{x^3 \left (-2+x^2\right )^2}\right ) \, dx \\ & = 2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} \left (-4-2 x+4 x^2+x^3\right )}{x^3 \left (-2+x^2\right )^2} \, dx \\ & = 2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx-\int \left (-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 x^2}+\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2}+\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \left (-2+x^2\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx-\frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{-2+x^2} \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx \\ & = \frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx-\frac {1}{2} \int \left (-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \sqrt {2} \left (\sqrt {2}-x\right )}-\frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{2 \sqrt {2} \left (\sqrt {2}+x\right )}\right ) \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx \\ & = \frac {1}{2} \int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^2} \, dx+2 \int e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \, dx+\frac {\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{\sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{\sqrt {2}+x} \, dx}{4 \sqrt {2}}+\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}}}{x^3} \, dx-\int \frac {e^{x+\frac {e^{-x}}{-2 x^2+x^4}} x}{\left (-2+x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{2 x+\frac {e^{-x}}{x^2 \left (-2+x^2\right )}} \]

[In]

Integrate[(E^(2*x + 1/(E^x*(-2*x^2 + x^4)))*(8*x^3 - 8*x^5 + 2*x^7 + (4 + 2*x - 4*x^2 - x^3)/E^x))/(4*x^3 - 4*
x^5 + x^7),x]

[Out]

E^(2*x + 1/(E^x*x^2*(-2 + x^2)))

Maple [A] (verified)

Time = 69.65 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08

method	result	size
risch	\({\mathrm e}^{\frac {2 x^{5}-4 x^{3}+{\mathrm e}^{-x}}{x^{2} \left (x^{2}-2\right )}}\)	\(28\)
parallelrisch	\(-\frac {\left (-24 \,{\mathrm e}^{2 x} x^{4}+48 \,{\mathrm e}^{2 x} x^{2}\right ) {\mathrm e}^{\frac {{\mathrm e}^{-x}}{x^{2} \left (x^{2}-2\right )}}}{24 x^{2} \left (x^{2}-2\right )}\)	\(51\)

[In]

int(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),x,meth
od=_RETURNVERBOSE)

[Out]

exp((2*x^5-4*x^3+exp(-x))/x^2/(x^2-2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{\left (\frac {2 \, x^{5} - 4 \, x^{3} + e^{\left (-x\right )}}{x^{4} - 2 \, x^{2}}\right )} \]

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="fricas")

[Out]

e^((2*x^5 - 4*x^3 + e^(-x))/(x^4 - 2*x^2))

Sympy [A] (verification not implemented)

Time = 2.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{2 x} e^{\frac {e^{- x}}{x^{4} - 2 x^{2}}} \]

[In]

integrate(((-x**3-4*x**2+2*x+4)*exp(-x)+2*x**7-8*x**5+8*x**3)*exp(2*x)/(x**7-4*x**5+4*x**3)/exp(-exp(-x)/(x**4
-2*x**2)),x)

[Out]

exp(2*x)*exp(exp(-x)/(x**4 - 2*x**2))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{\left (2 \, x + \frac {e^{\left (-x\right )}}{2 \, {\left (x^{2} - 2\right )}} - \frac {e^{\left (-x\right )}}{2 \, x^{2}}\right )} \]

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx=e^{\left (\frac {2 \, x^{5} - 4 \, x^{3} + e^{\left (-x\right )}}{x^{4} - 2 \, x^{2}}\right )} \]

[In]

integrate(((-x^3-4*x^2+2*x+4)*exp(-x)+2*x^7-8*x^5+8*x^3)*exp(2*x)/(x^7-4*x^5+4*x^3)/exp(-exp(-x)/(x^4-2*x^2)),
x, algorithm="giac")

[Out]

e^((2*x^5 - 4*x^3 + e^(-x))/(x^4 - 2*x^2))

Mupad [B] (verification not implemented)

Time = 13.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {e^{2 x+\frac {e^{-x}}{-2 x^2+x^4}} \left (8 x^3-8 x^5+2 x^7+e^{-x} \left (4+2 x-4 x^2-x^3\right )\right )}{4 x^3-4 x^5+x^7} \, dx={\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x}}{2\,x^2-x^4}}\,{\mathrm {e}}^{2\,x} \]

[In]

int((exp(-exp(-x)/(2*x^2 - x^4))*exp(2*x)*(exp(-x)*(2*x - 4*x^2 - x^3 + 4) + 8*x^3 - 8*x^5 + 2*x^7))/(4*x^3 -
4*x^5 + x^7),x)

[Out]

exp(-exp(-x)/(2*x^2 - x^4))*exp(2*x)