3.5 \(\int \frac{e^{\frac{1}{-1+x^2}} (1-3 x-x^2+x^3)}{1-x-x^2+x^3} \, dx\)

Optimal. Leaf size=13 \[ e^{\frac{1}{x^2-1}} (x+1) \]

[Out]

E^(-1 + x^2)^(-1)*(1 + x)

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Rubi [F]  time = 0.388562, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{\frac{1}{-1+x^2}} \left (1-3 x-x^2+x^3\right )}{1-x-x^2+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{e^{\frac{1}{-1+x^2}} \left (1-3 x-x^2+x^3\right )}{1-x-x^2+x^3} \, dx &=\int \left (e^{\frac{1}{-1+x^2}}-\frac{2 e^{\frac{1}{-1+x^2}} x}{1-x-x^2+x^3}\right ) \, dx\\ &=-\left (2 \int \frac{e^{\frac{1}{-1+x^2}} x}{1-x-x^2+x^3} \, dx\right )+\int e^{\frac{1}{-1+x^2}} \, dx\\ &=-\left (2 \int \left (\frac{e^{\frac{1}{-1+x^2}}}{2 (-1+x)^2}+\frac{e^{\frac{1}{-1+x^2}}}{2 \left (-1+x^2\right )}\right ) \, dx\right )+\int e^{\frac{1}{-1+x^2}} \, dx\\ &=\int e^{\frac{1}{-1+x^2}} \, dx-\int \frac{e^{\frac{1}{-1+x^2}}}{(-1+x)^2} \, dx-\int \frac{e^{\frac{1}{-1+x^2}}}{-1+x^2} \, dx\\ &=\int e^{\frac{1}{-1+x^2}} \, dx-\int \frac{e^{\frac{1}{-1+x^2}}}{(-1+x)^2} \, dx-\int \left (-\frac{e^{\frac{1}{-1+x^2}}}{2 (1-x)}-\frac{e^{\frac{1}{-1+x^2}}}{2 (1+x)}\right ) \, dx\\ &=\frac{1}{2} \int \frac{e^{\frac{1}{-1+x^2}}}{1-x} \, dx+\frac{1}{2} \int \frac{e^{\frac{1}{-1+x^2}}}{1+x} \, dx+\int e^{\frac{1}{-1+x^2}} \, dx-\int \frac{e^{\frac{1}{-1+x^2}}}{(-1+x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.216282, size = 13, normalized size = 1. \[ e^{\frac{1}{x^2-1}} (x+1) \]

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-1 + x^2)^(-1)*(1 + x)

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Maple [A]  time = 0.003, size = 13, normalized size = 1. \begin{align*}{{\rm e}^{ \left ({x}^{2}-1 \right ) ^{-1}}} \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x^{3} - x^{2} - 3 \, x + 1\right )} e^{\left (\frac{1}{x^{2} - 1}\right )}}{x^{3} - x^{2} - x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x^2-3*x+1)*exp(1/(x^2-1))/(x^3-x^2-x+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.50929, size = 34, normalized size = 2.62 \begin{align*}{\left (x + 1\right )} e^{\left (\frac{1}{x^{2} - 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-x^2-3*x+1)*exp(1/(x^2-1))/(x^3-x^2-x+1),x, algorithm="fricas")

[Out]

(x + 1)*e^(1/(x^2 - 1))

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Sympy [A]  time = 0.15851, size = 10, normalized size = 0.77 \begin{align*} \left (x + 1\right ) e^{\frac{1}{x^{2} - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-x**2-3*x+1)*exp(1/(x**2-1))/(x**3-x**2-x+1),x)

[Out]

(x + 1)*exp(1/(x**2 - 1))

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Giac [B]  time = 1.08291, size = 41, normalized size = 3.15 \begin{align*}{\left (x e^{\left (\frac{x^{2}}{x^{2} - 1}\right )} + e^{\left (\frac{x^{2}}{x^{2} - 1}\right )}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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