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

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

[Out]

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

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Rubi [F]  time = 0.41, 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 = {} \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx \]

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*}

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Mathematica [A]  time = 0.11, size = 10, normalized size = 1.00 \[ \operatorname {ExpIntegralEi}\left (\frac {x}{x^2+2}\right ) \]

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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fricas [A]  time = 0.43, size = 10, normalized size = 1.00 \[ {\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \]

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

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)

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maple [F]  time = 0.10, 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 \[ -\int \frac {{\left (x^{2} - 2\right )} e^{\left (\frac {x}{x^{2} + 2}\right )}}{x^{3} + 2 \, x}\,{d x} \]

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

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

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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