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

3.1.2.1 Optimal result

Integrand size = 28, antiderivative size = 10 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx=\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \]

output

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

3.1.2.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx=\operatorname {ExpIntegralEi}\left (\frac {x}{2+x^2}\right ) \]

input

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

output

ExpIntegralEi[x/(2 + x^2)]
 

3.1.2.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{\frac {x}{x^2+2}} \left (2-x^2\right )}{x \left (x^2+2\right )}dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (\frac {e^{\frac {x}{x^2+2}}}{x}-\frac {2 e^{\frac {x}{x^2+2}} x}{x^2+2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {e^{\frac {x}{x^2+2}}}{i \sqrt {2}-x}dx+\int \frac {e^{\frac {x}{x^2+2}}}{x}dx-\int \frac {e^{\frac {x}{x^2+2}}}{x+i \sqrt {2}}dx\)

input

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

output

$Aborted
 

3.1.2.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.1.2.4 Maple [F]

\[\int \frac {\left (-x^{2}+2\right ) {\mathrm e}^{\frac {x}{x^{2}+2}}}{x^{3}+2 x}d x\]

input

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

output

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

3.1.2.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx={\rm Ei}\left (\frac {x}{x^{2} + 2}\right ) \]

input

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

output

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

3.1.2.6 Sympy [F]

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

input

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

output

-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)
 

3.1.2.7 Maxima [F]

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

input

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

output

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

3.1.2.8 Giac [F]

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

input

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

output

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

3.1.2.9 Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx=\mathrm {ei}\left (\frac {x}{x^2+2}\right ) \]

input

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

output

ei(x/(x^2 + 2))