3.50.84 \(\int \frac {-5 x^2+e^{e^{\frac {4}{x^2}}} (e^{\frac {4}{x^2}} (120-24 x)-15 x^2)}{x^4+6 e^{e^{\frac {4}{x^2}}} x^4+9 e^{2 e^{\frac {4}{x^2}}} x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {5-x}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right ) x} \]

________________________________________________________________________________________

Rubi [F]  time = 1.76, 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 {-5 x^2+e^{e^{\frac {4}{x^2}}} \left (e^{\frac {4}{x^2}} (120-24 x)-15 x^2\right )}{x^4+6 e^{e^{\frac {4}{x^2}}} x^4+9 e^{2 e^{\frac {4}{x^2}}} x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x^2 + E^E^(4/x^2)*(E^(4/x^2)*(120 - 24*x) - 15*x^2))/(x^4 + 6*E^E^(4/x^2)*x^4 + 9*E^(2*E^(4/x^2))*x^4)
,x]

[Out]

-(1 + 3*E^E^(4/x^2))^(-1) + 120*Defer[Int][E^(E^(4/x^2) + 4/x^2)/((1 + 3*E^E^(4/x^2))^2*x^4), x] + 5*Defer[Sub
st][Defer[Int][(1 + 3*E^E^(4*x^2))^(-1), x], x, x^(-1)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-5 x^2+e^{e^{\frac {4}{x^2}}} \left (e^{\frac {4}{x^2}} (120-24 x)-15 x^2\right )}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4} \, dx\\ &=\int \left (-\frac {24 e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}} (-5+x)}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4}-\frac {5}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right ) x^2}\right ) \, dx\\ &=-\left (5 \int \frac {1}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right ) x^2} \, dx\right )-24 \int \frac {e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}} (-5+x)}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4} \, dx\\ &=5 \operatorname {Subst}\left (\int \frac {1}{1+3 e^{e^{4 x^2}}} \, dx,x,\frac {1}{x}\right )-24 \int \left (-\frac {5 e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}}}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4}+\frac {e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}}}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^3}\right ) \, dx\\ &=5 \operatorname {Subst}\left (\int \frac {1}{1+3 e^{e^{4 x^2}}} \, dx,x,\frac {1}{x}\right )-24 \int \frac {e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}}}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^3} \, dx+120 \int \frac {e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}}}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4} \, dx\\ &=-\frac {1}{1+3 e^{e^{\frac {4}{x^2}}}}+5 \operatorname {Subst}\left (\int \frac {1}{1+3 e^{e^{4 x^2}}} \, dx,x,\frac {1}{x}\right )+120 \int \frac {e^{e^{\frac {4}{x^2}}+\frac {4}{x^2}}}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right )^2 x^4} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 23, normalized size = 0.96 \begin {gather*} -\frac {-5+x}{\left (1+3 e^{e^{\frac {4}{x^2}}}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x^2 + E^E^(4/x^2)*(E^(4/x^2)*(120 - 24*x) - 15*x^2))/(x^4 + 6*E^E^(4/x^2)*x^4 + 9*E^(2*E^(4/x^2)
)*x^4),x]

[Out]

-((-5 + x)/((1 + 3*E^E^(4/x^2))*x))

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 19, normalized size = 0.79 \begin {gather*} -\frac {x - 5}{3 \, x e^{\left (e^{\left (\frac {4}{x^{2}}\right )}\right )} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x+120)*exp(4/x^2)-15*x^2)*exp(exp(4/x^2))-5*x^2)/(9*x^4*exp(exp(4/x^2))^2+6*x^4*exp(exp(4/x^2
))+x^4),x, algorithm="fricas")

[Out]

-(x - 5)/(3*x*e^(e^(4/x^2)) + x)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 19, normalized size = 0.79 \begin {gather*} -\frac {x - 5}{3 \, x e^{\left (e^{\left (\frac {4}{x^{2}}\right )}\right )} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x+120)*exp(4/x^2)-15*x^2)*exp(exp(4/x^2))-5*x^2)/(9*x^4*exp(exp(4/x^2))^2+6*x^4*exp(exp(4/x^2
))+x^4),x, algorithm="giac")

[Out]

-(x - 5)/(3*x*e^(e^(4/x^2)) + x)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 22, normalized size = 0.92

method	result	size
risch	\(-\frac {x -5}{x \left (1+3 \,{\mathrm e}^{{\mathrm e}^{\frac {4}{x^{2}}}}\right )}\)	\(22\)
norman	\(\frac {3 x^{3} {\mathrm e}^{{\mathrm e}^{\frac {4}{x^{2}}}}+5 x^{2}}{x^{3} \left (1+3 \,{\mathrm e}^{{\mathrm e}^{\frac {4}{x^{2}}}}\right )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-24*x+120)*exp(4/x^2)-15*x^2)*exp(exp(4/x^2))-5*x^2)/(9*x^4*exp(exp(4/x^2))^2+6*x^4*exp(exp(4/x^2))+x^4
),x,method=_RETURNVERBOSE)

[Out]

-(x-5)/x/(1+3*exp(exp(4/x^2)))

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 19, normalized size = 0.79 \begin {gather*} -\frac {x - 5}{3 \, x e^{\left (e^{\left (\frac {4}{x^{2}}\right )}\right )} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x+120)*exp(4/x^2)-15*x^2)*exp(exp(4/x^2))-5*x^2)/(9*x^4*exp(exp(4/x^2))^2+6*x^4*exp(exp(4/x^2
))+x^4),x, algorithm="maxima")

[Out]

-(x - 5)/(3*x*e^(e^(4/x^2)) + x)

________________________________________________________________________________________

mupad [B]  time = 3.52, size = 21, normalized size = 0.88 \begin {gather*} -\frac {x-5}{x\,\left (3\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {4}{x^2}}}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(4/x^2))*(exp(4/x^2)*(24*x - 120) + 15*x^2) + 5*x^2)/(6*x^4*exp(exp(4/x^2)) + 9*x^4*exp(2*exp(4/x
^2)) + x^4),x)

[Out]

-(x - 5)/(x*(3*exp(exp(4/x^2)) + 1))

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 15, normalized size = 0.62 \begin {gather*} \frac {5 - x}{3 x e^{e^{\frac {4}{x^{2}}}} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-24*x+120)*exp(4/x**2)-15*x**2)*exp(exp(4/x**2))-5*x**2)/(9*x**4*exp(exp(4/x**2))**2+6*x**4*exp(e
xp(4/x**2))+x**4),x)

[Out]

(5 - x)/(3*x*exp(exp(4/x**2)) + x)

________________________________________________________________________________________