\(\int \frac {4 x^3+e^{e^{x^2}} (2-4 e^{x^2} x^2)}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx\) [5501]

Optimal result

Integrand size = 54, antiderivative size = 24 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=2 \left (4+e^5+\frac {x}{e^{e^{x^2}}-x^3}\right ) \]

[Out]

8+2*x/(exp(exp(x^2))-x^3)+2*exp(5)

Rubi [F]

\[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=\int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx \]

[In]

Int[(4*x^3 + E^E^x^2*(2 - 4*E^x^2*x^2))/(E^(2*E^x^2) - 2*E^E^x^2*x^3 + x^6),x]

[Out]

-4*Defer[Int][(E^(E^x^2 + x^2)*x^2)/(E^E^x^2 - x^3)^2, x] + 2*Defer[Int][(E^E^x^2 - x^3)^(-1), x] + 6*Defer[In
t][x^3/(-E^E^x^2 + x^3)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{\left (e^{e^{x^2}}-x^3\right )^2} \, dx \\ & = \int \left (-\frac {4 e^{e^{x^2}+x^2} x^2}{\left (e^{e^{x^2}}-x^3\right )^2}+\frac {2 \left (e^{e^{x^2}}+2 x^3\right )}{\left (e^{e^{x^2}}-x^3\right )^2}\right ) \, dx \\ & = 2 \int \frac {e^{e^{x^2}}+2 x^3}{\left (e^{e^{x^2}}-x^3\right )^2} \, dx-4 \int \frac {e^{e^{x^2}+x^2} x^2}{\left (e^{e^{x^2}}-x^3\right )^2} \, dx \\ & = 2 \int \left (\frac {1}{e^{e^{x^2}}-x^3}+\frac {3 x^3}{\left (-e^{e^{x^2}}+x^3\right )^2}\right ) \, dx-4 \int \frac {e^{e^{x^2}+x^2} x^2}{\left (e^{e^{x^2}}-x^3\right )^2} \, dx \\ & = 2 \int \frac {1}{e^{e^{x^2}}-x^3} \, dx-4 \int \frac {e^{e^{x^2}+x^2} x^2}{\left (e^{e^{x^2}}-x^3\right )^2} \, dx+6 \int \frac {x^3}{\left (-e^{e^{x^2}}+x^3\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.75 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=\frac {2 x}{e^{e^{x^2}}-x^3} \]

[In]

Integrate[(4*x^3 + E^E^x^2*(2 - 4*E^x^2*x^2))/(E^(2*E^x^2) - 2*E^E^x^2*x^3 + x^6),x]

[Out]

(2*x)/(E^E^x^2 - x^3)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {2 x}{x^{3}-{\mathrm e}^{{\mathrm e}^{x^{2}}}}\)	\(17\)
parallelrisch	\(-\frac {2 x}{x^{3}-{\mathrm e}^{{\mathrm e}^{x^{2}}}}\)	\(17\)

[In]

int(((-4*x^2*exp(x^2)+2)*exp(exp(x^2))+4*x^3)/(exp(exp(x^2))^2-2*x^3*exp(exp(x^2))+x^6),x,method=_RETURNVERBOS
E)

[Out]

-2*x/(x^3-exp(exp(x^2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=-\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \]

[In]

integrate(((-4*x^2*exp(x^2)+2)*exp(exp(x^2))+4*x^3)/(exp(exp(x^2))^2-2*x^3*exp(exp(x^2))+x^6),x, algorithm="fr
icas")

[Out]

-2*x/(x^3 - e^(e^(x^2)))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.50 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=\frac {2 x}{- x^{3} + e^{e^{x^{2}}}} \]

[In]

integrate(((-4*x**2*exp(x**2)+2)*exp(exp(x**2))+4*x**3)/(exp(exp(x**2))**2-2*x**3*exp(exp(x**2))+x**6),x)

[Out]

2*x/(-x**3 + exp(exp(x**2)))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=-\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \]

[In]

integrate(((-4*x^2*exp(x^2)+2)*exp(exp(x^2))+4*x^3)/(exp(exp(x^2))^2-2*x^3*exp(exp(x^2))+x^6),x, algorithm="ma
xima")

[Out]

-2*x/(x^3 - e^(e^(x^2)))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=-\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \]

[In]

integrate(((-4*x^2*exp(x^2)+2)*exp(exp(x^2))+4*x^3)/(exp(exp(x^2))^2-2*x^3*exp(exp(x^2))+x^6),x, algorithm="gi
ac")

[Out]

-2*x/(x^3 - e^(e^(x^2)))

Mupad [B] (verification not implemented)

Time = 13.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {4 x^3+e^{e^{x^2}} \left (2-4 e^{x^2} x^2\right )}{e^{2 e^{x^2}}-2 e^{e^{x^2}} x^3+x^6} \, dx=\frac {2\,x}{{\mathrm {e}}^{{\mathrm {e}}^{x^2}}-x^3} \]

[In]

int(-(exp(exp(x^2))*(4*x^2*exp(x^2) - 2) - 4*x^3)/(exp(2*exp(x^2)) + x^6 - 2*x^3*exp(exp(x^2))),x)

[Out]

(2*x)/(exp(exp(x^2)) - x^3)