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

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

[Out]

Ei(x/(x^2+2))

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Rubi [F]
time = 0.23, 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 {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.07, size = 10, normalized size = 1.00 \begin {gather*} \text {Ei}\left (\frac {x}{2+x^2}\right ) \end {gather*}

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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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(2 - x^2)*Exp[x/(x^2 + 2)]/(x^3 + 2*x),x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (-x^{2}+2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.30, size = 10, normalized size = 1.00 \begin {gather*} {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {2 e^{\frac {x}{x^{2} + 2}}}{x^{3} + 2 x}\right )\, dx - \int \frac {x^{2} e^{\frac {x}{x^{2} + 2}}}{x^{3} + 2 x}\, dx \end {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

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]

Could not integrate

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Mupad [B]
time = 0.24, size = 10, normalized size = 1.00 \begin {gather*} \mathrm {ei}\left (\frac {x}{x^2+2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ei(x/(x^2 + 2))

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