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

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))
 

Mathematica [A] (verified)

Time = 0.11 (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)]
 

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
 

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]
 

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)
 

Fricas [A] (verification not implemented)

Time = 0.07 (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))
 

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)
 

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)
 

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)
 

Mupad [B] (verification not implemented)

Time = 0.15 (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))
 

Reduce [F]

\[ \int \frac {e^{\frac {x}{2+x^2}} \left (2-x^2\right )}{2 x+x^3} \, dx=\frac {2 e^{\frac {x}{x^{2}+2}} x^{4}-2 e^{\frac {x}{x^{2}+2}} x^{3}-4 e^{\frac {x}{x^{2}+2}} x^{2}-32 e^{\frac {x}{x^{2}+2}}+208 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{9}+8 x^{7}+24 x^{5}+32 x^{3}+16 x}d x \right ) x^{4}+832 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{9}+8 x^{7}+24 x^{5}+32 x^{3}+16 x}d x \right ) x^{2}+832 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{9}+8 x^{7}+24 x^{5}+32 x^{3}+16 x}d x \right )+64 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right ) x^{4}+256 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right ) x^{2}+256 \left (\int \frac {e^{\frac {x}{x^{2}+2}}}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right )+88 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right ) x^{4}+352 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right ) x^{2}+352 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{8}+8 x^{6}+24 x^{4}+32 x^{2}+16}d x \right )-13 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{2}+2}d x \right ) x^{4}-52 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{2}+2}d x \right ) x^{2}-52 \left (\int \frac {e^{\frac {x}{x^{2}+2}} x}{x^{2}+2}d x \right )}{13 x^{4}+52 x^{2}+52} \] Input:

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

Output:

(2*e**(x/(x**2 + 2))*x**4 - 2*e**(x/(x**2 + 2))*x**3 - 4*e**(x/(x**2 + 2)) 
*x**2 - 32*e**(x/(x**2 + 2)) + 208*int(e**(x/(x**2 + 2))/(x**9 + 8*x**7 + 
24*x**5 + 32*x**3 + 16*x),x)*x**4 + 832*int(e**(x/(x**2 + 2))/(x**9 + 8*x* 
*7 + 24*x**5 + 32*x**3 + 16*x),x)*x**2 + 832*int(e**(x/(x**2 + 2))/(x**9 + 
 8*x**7 + 24*x**5 + 32*x**3 + 16*x),x) + 64*int(e**(x/(x**2 + 2))/(x**8 + 
8*x**6 + 24*x**4 + 32*x**2 + 16),x)*x**4 + 256*int(e**(x/(x**2 + 2))/(x**8 
 + 8*x**6 + 24*x**4 + 32*x**2 + 16),x)*x**2 + 256*int(e**(x/(x**2 + 2))/(x 
**8 + 8*x**6 + 24*x**4 + 32*x**2 + 16),x) + 88*int((e**(x/(x**2 + 2))*x)/( 
x**8 + 8*x**6 + 24*x**4 + 32*x**2 + 16),x)*x**4 + 352*int((e**(x/(x**2 + 2 
))*x)/(x**8 + 8*x**6 + 24*x**4 + 32*x**2 + 16),x)*x**2 + 352*int((e**(x/(x 
**2 + 2))*x)/(x**8 + 8*x**6 + 24*x**4 + 32*x**2 + 16),x) - 13*int((e**(x/( 
x**2 + 2))*x)/(x**2 + 2),x)*x**4 - 52*int((e**(x/(x**2 + 2))*x)/(x**2 + 2) 
,x)*x**2 - 52*int((e**(x/(x**2 + 2))*x)/(x**2 + 2),x))/(13*(x**4 + 4*x**2 
+ 4))