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

Optimal. Leaf size=24 \[ 2 \left (4+e^5+\frac {x}{e^{e^{x^2}}-x^3}\right ) \]

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Rubi [F]  time = 0.84, 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*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 18, normalized size = 0.75 \begin {gather*} \frac {2 x}{e^{e^{x^2}}-x^3} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.60, size = 16, normalized size = 0.67 \begin {gather*} -\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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giac [A]  time = 0.17, size = 16, normalized size = 0.67 \begin {gather*} -\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maple [A]  time = 0.04, size = 17, normalized size = 0.71

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

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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maxima [A]  time = 0.41, size = 16, normalized size = 0.67 \begin {gather*} -\frac {2 \, x}{x^{3} - e^{\left (e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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mupad [B]  time = 3.86, size = 16, normalized size = 0.67 \begin {gather*} \frac {2\,x}{{\mathrm {e}}^{{\mathrm {e}}^{x^2}}-x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.23, size = 12, normalized size = 0.50 \begin {gather*} \frac {2 x}{- x^{3} + e^{e^{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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