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

Optimal. Leaf size=20 \[ e^{-4+(2+x)^2+\frac {2}{1-x^2}} \]

________________________________________________________________________________________

Rubi [F]  time = 1.52, 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^{\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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.29, size = 18, normalized size = 0.90 \begin {gather*} e^{4 x+x^2-\frac {2}{-1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[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))

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 27, normalized size = 1.35 \begin {gather*} e^{\left (\frac {x^{4} + 4 \, x^{3} - x^{2} - 4 \, x - 2}{x^{2} - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [B]  time = 0.36, size = 56, normalized size = 2.80 \begin {gather*} 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.17, size = 28, normalized size = 1.40

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maxima [A]  time = 0.65, size = 20, normalized size = 1.00 \begin {gather*} e^{\left (x^{2} + 4 \, x + \frac {1}{x + 1} - \frac {1}{x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

mupad [B]  time = 0.74, size = 60, normalized size = 3.00 \begin {gather*} {\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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

sympy [A]  time = 0.22, size = 22, normalized size = 1.10 \begin {gather*} e^{\frac {x^{4} + 4 x^{3} - x^{2} - 4 x - 2}{x^{2} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________