3.103.41 \(\int \frac {e^{e^x+x^2} (4+2 x-8 x^2-2 x^3+e^x (-4 x-x^2))+e^{x^2} (-x^2+8 x^3+2 x^4+e^x (32-16 x-78 x^2-34 x^3-4 x^4))}{16 x^2+8 x^3+x^4} \, dx\)

Optimal. Leaf size=34 \[ \frac {2 e^{x^2} \left (-e^x+\frac {-e^{e^x}+x}{2 (4+x)}\right )}{x} \]

________________________________________________________________________________________

Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^x + x^2)*(4 + 2*x - 8*x^2 - 2*x^3 + E^x*(-4*x - x^2)) + E^x^2*(-x^2 + 8*x^3 + 2*x^4 + E^x*(32 - 16*x
- 78*x^2 - 34*x^3 - 4*x^4)))/(16*x^2 + 8*x^3 + x^4),x]

[Out]

$Aborted

Rubi steps

Aborted

________________________________________________________________________________________

Mathematica [A]  time = 3.63, size = 32, normalized size = 0.94 \begin {gather*} -\frac {e^{x^2} \left (e^{e^x}-x+2 e^x (4+x)\right )}{x (4+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^x + x^2)*(4 + 2*x - 8*x^2 - 2*x^3 + E^x*(-4*x - x^2)) + E^x^2*(-x^2 + 8*x^3 + 2*x^4 + E^x*(32
- 16*x - 78*x^2 - 34*x^3 - 4*x^4)))/(16*x^2 + 8*x^3 + x^4),x]

[Out]

-((E^x^2*(E^E^x - x + 2*E^x*(4 + x)))/(x*(4 + x)))

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 35, normalized size = 1.03 \begin {gather*} -\frac {{\left (2 \, {\left (x + 4\right )} e^{x} - x\right )} e^{\left (x^{2}\right )} + e^{\left (x^{2} + e^{x}\right )}}{x^{2} + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-4*x)*exp(x)-2*x^3-8*x^2+2*x+4)*exp(x^2)*exp(exp(x))+((-4*x^4-34*x^3-78*x^2-16*x+32)*exp(x)+2
*x^4+8*x^3-x^2)*exp(x^2))/(x^4+8*x^3+16*x^2),x, algorithm="fricas")

[Out]

-((2*(x + 4)*e^x - x)*e^(x^2) + e^(x^2 + e^x))/(x^2 + 4*x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{3} + 8 \, x^{2} + {\left (x^{2} + 4 \, x\right )} e^{x} - 2 \, x - 4\right )} e^{\left (x^{2} + e^{x}\right )} - {\left (2 \, x^{4} + 8 \, x^{3} - x^{2} - 2 \, {\left (2 \, x^{4} + 17 \, x^{3} + 39 \, x^{2} + 8 \, x - 16\right )} e^{x}\right )} e^{\left (x^{2}\right )}}{x^{4} + 8 \, x^{3} + 16 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-4*x)*exp(x)-2*x^3-8*x^2+2*x+4)*exp(x^2)*exp(exp(x))+((-4*x^4-34*x^3-78*x^2-16*x+32)*exp(x)+2
*x^4+8*x^3-x^2)*exp(x^2))/(x^4+8*x^3+16*x^2),x, algorithm="giac")

[Out]

integrate(-((2*x^3 + 8*x^2 + (x^2 + 4*x)*e^x - 2*x - 4)*e^(x^2 + e^x) - (2*x^4 + 8*x^3 - x^2 - 2*(2*x^4 + 17*x
^3 + 39*x^2 + 8*x - 16)*e^x)*e^(x^2))/(x^4 + 8*x^3 + 16*x^2), x)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 46, normalized size = 1.35

method	result	size
risch	\(-\frac {\left (2 \,{\mathrm e}^{x} x -x +8 \,{\mathrm e}^{x}\right ) {\mathrm e}^{x^{2}}}{\left (4+x \right ) x}-\frac {{\mathrm e}^{x^{2}+{\mathrm e}^{x}}}{x \left (4+x \right )}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-4*x)*exp(x)-2*x^3-8*x^2+2*x+4)*exp(x^2)*exp(exp(x))+((-4*x^4-34*x^3-78*x^2-16*x+32)*exp(x)+2*x^4+8
*x^3-x^2)*exp(x^2))/(x^4+8*x^3+16*x^2),x,method=_RETURNVERBOSE)

[Out]

-(2*exp(x)*x-x+8*exp(x))/(4+x)/x*exp(x^2)-1/x/(4+x)*exp(x^2+exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 35, normalized size = 1.03 \begin {gather*} -\frac {{\left (2 \, {\left (x + 4\right )} e^{x} - x\right )} e^{\left (x^{2}\right )} + e^{\left (x^{2} + e^{x}\right )}}{x^{2} + 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-4*x)*exp(x)-2*x^3-8*x^2+2*x+4)*exp(x^2)*exp(exp(x))+((-4*x^4-34*x^3-78*x^2-16*x+32)*exp(x)+2
*x^4+8*x^3-x^2)*exp(x^2))/(x^4+8*x^3+16*x^2),x, algorithm="maxima")

[Out]

-((2*(x + 4)*e^x - x)*e^(x^2) + e^(x^2 + e^x))/(x^2 + 4*x)

________________________________________________________________________________________

mupad [B]  time = 0.42, size = 47, normalized size = 1.38 \begin {gather*} -\frac {{\mathrm {e}}^{{\mathrm {e}}^x+x^2}}{x^2+4\,x}-\frac {{\mathrm {e}}^{x^2}\,\left (8\,{\mathrm {e}}^x-x+2\,x\,{\mathrm {e}}^x\right )}{x^2+4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x^2)*(exp(x)*(16*x + 78*x^2 + 34*x^3 + 4*x^4 - 32) + x^2 - 8*x^3 - 2*x^4) + exp(x^2)*exp(exp(x))*(ex
p(x)*(4*x + x^2) - 2*x + 8*x^2 + 2*x^3 - 4))/(16*x^2 + 8*x^3 + x^4),x)

[Out]

- exp(exp(x) + x^2)/(4*x + x^2) - (exp(x^2)*(8*exp(x) - x + 2*x*exp(x)))/(4*x + x^2)

________________________________________________________________________________________

sympy [A]  time = 0.36, size = 41, normalized size = 1.21 \begin {gather*} \frac {\left (- 2 x e^{x} + x - 8 e^{x}\right ) e^{x^{2}}}{x^{2} + 4 x} - \frac {e^{x^{2}} e^{e^{x}}}{x^{2} + 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-4*x)*exp(x)-2*x**3-8*x**2+2*x+4)*exp(x**2)*exp(exp(x))+((-4*x**4-34*x**3-78*x**2-16*x+32)*e
xp(x)+2*x**4+8*x**3-x**2)*exp(x**2))/(x**4+8*x**3+16*x**2),x)

[Out]

(-2*x*exp(x) + x - 8*exp(x))*exp(x**2)/(x**2 + 4*x) - exp(x**2)*exp(exp(x))/(x**2 + 4*x)

________________________________________________________________________________________