\(\int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5)}{1-2 x^2+x^4} \, dx\) [658]

Optimal result

Integrand size = 66, antiderivative size = 20 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{-4+(2+x)^2+\frac {2}{1-x^2}} \]

[Out]

exp(2/(-x^2+1)+(2+x)^2-4)

Rubi [F]

\[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=\int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx \]

[In]

Int[(E^((-2 - 4*x - x^2 + 4*x^3 + x^4)/(-1 + x^2))*(4 + 6*x - 8*x^2 - 4*x^3 + 4*x^4 + 2*x^5))/(1 - 2*x^2 + x^4
),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 1.47 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{4 x+x^2-\frac {2}{-1+x^2}} \]

[In]

Integrate[(E^((-2 - 4*x - x^2 + 4*x^3 + x^4)/(-1 + x^2))*(4 + 6*x - 8*x^2 - 4*x^3 + 4*x^4 + 2*x^5))/(1 - 2*x^2
+ x^4),x]

[Out]

E^(4*x + x^2 - 2/(-1 + x^2))

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40

method	result	size
gosper	\({\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}\)	\(28\)
parallelrisch	\({\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}\)	\(28\)
risch	\({\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{\left (-1+x \right ) \left (1+x \right )}}\)	\(31\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}-{\mathrm e}^{\frac {x^{4}+4 x^{3}-x^{2}-4 x -2}{x^{2}-1}}}{x^{2}-1}\)	\(70\)

[In]

int((2*x^5+4*x^4-4*x^3-8*x^2+6*x+4)*exp((x^4+4*x^3-x^2-4*x-2)/(x^2-1))/(x^4-2*x^2+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{\left (\frac {x^{4} + 4 \, x^{3} - x^{2} - 4 \, x - 2}{x^{2} - 1}\right )} \]

[In]

integrate((2*x^5+4*x^4-4*x^3-8*x^2+6*x+4)*exp((x^4+4*x^3-x^2-4*x-2)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="frica
s")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{\frac {x^{4} + 4 x^{3} - x^{2} - 4 x - 2}{x^{2} - 1}} \]

[In]

integrate((2*x**5+4*x**4-4*x**3-8*x**2+6*x+4)*exp((x**4+4*x**3-x**2-4*x-2)/(x**2-1))/(x**4-2*x**2+1),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{\left (x^{2} + 4 \, x + \frac {1}{x + 1} - \frac {1}{x - 1}\right )} \]

[In]

integrate((2*x^5+4*x^4-4*x^3-8*x^2+6*x+4)*exp((x^4+4*x^3-x^2-4*x-2)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="maxim
a")

[Out]

e^(x^2 + 4*x + 1/(x + 1) - 1/(x - 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.80 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx=e^{\left (\frac {x^{4}}{x^{2} - 1} + \frac {4 \, x^{3}}{x^{2} - 1} - \frac {x^{2}}{x^{2} - 1} - \frac {4 \, x}{x^{2} - 1} - \frac {2}{x^{2} - 1}\right )} \]

[In]

integrate((2*x^5+4*x^4-4*x^3-8*x^2+6*x+4)*exp((x^4+4*x^3-x^2-4*x-2)/(x^2-1))/(x^4-2*x^2+1),x, algorithm="giac"
)

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.00 \[ \int \frac {e^{\frac {-2-4 x-x^2+4 x^3+x^4}{-1+x^2}} \left (4+6 x-8 x^2-4 x^3+4 x^4+2 x^5\right )}{1-2 x^2+x^4} \, dx={\mathrm {e}}^{-\frac {x^2}{x^2-1}}\,{\mathrm {e}}^{\frac {x^4}{x^2-1}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^2-1}}\,{\mathrm {e}}^{-\frac {2}{x^2-1}}\,{\mathrm {e}}^{-\frac {4\,x}{x^2-1}} \]

[In]

int((exp(-(4*x + x^2 - 4*x^3 - x^4 + 2)/(x^2 - 1))*(6*x - 8*x^2 - 4*x^3 + 4*x^4 + 2*x^5 + 4))/(x^4 - 2*x^2 + 1
),x)

[Out]

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