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

Optimal. Leaf size=10 \[ \text{ExpIntegralEi}\left (\frac{x}{x^2+2}\right ) \]

[Out]

ExpIntegralEi[x/(2 + x^2)]

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Rubi [F]  time = 0.408486, 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 = {} \[ \int \frac{e^{\frac{x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0994643, size = 10, normalized size = 1. \[ \text{ExpIntegralEi}\left (\frac{x}{x^2+2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ExpIntegralEi[x/(2 + x^2)]

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{-{x}^{2}+2}{{x}^{3}+2\,x}{{\rm e}^{{\frac{x}{{x}^{2}+2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50984, size = 23, normalized size = 2.3 \begin{align*}{\rm Ei}\left (\frac{x}{x^{2} + 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ei(x/(x^2 + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{2 e^{\frac{x}{x^{2} + 2}}}{x^{3} + 2 x}\, dx - \int \frac{x^{2} e^{\frac{x}{x^{2} + 2}}}{x^{3} + 2 x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-2*exp(x/(x**2 + 2))/(x**3 + 2*x), x) - Integral(x**2*exp(x/(x**2 + 2))/(x**3 + 2*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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