\(\int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} (144 e^{\frac {3}{x^2}}-36 x^2)}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx\) [505]

Optimal result

Integrand size = 91, antiderivative size = 23 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {x^3}{\left (12 e^{-e^{\frac {3}{x^2}}}-x\right )^2} \]

[Out]

x^3/(12/exp(exp(3/x^2))-x)^2

Rubi [F]

\[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx \]

[In]

Int[(E^(3*E^(3/x^2))*x^3 + E^(2*E^(3/x^2))*(144*E^(3/x^2) - 36*x^2))/(-1728 + 432*E^E^(3/x^2)*x - 36*E^(2*E^(3
/x^2))*x^2 + E^(3*E^(3/x^2))*x^3),x]

[Out]

144*Defer[Int][E^(2*E^(3/x^2) + 3/x^2)/(-12 + E^E^(3/x^2)*x)^3, x] - 24*Defer[Int][(E^(2*E^(3/x^2))*x^2)/(-12
+ E^E^(3/x^2)*x)^3, x] + Defer[Int][(E^(2*E^(3/x^2))*x^2)/(-12 + E^E^(3/x^2)*x)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{2 e^{\frac {3}{x^2}}} \left (-144 e^{\frac {3}{x^2}}+36 x^2-e^{e^{\frac {3}{x^2}}} x^3\right )}{\left (12-e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx \\ & = \int \left (\frac {144 e^{2 e^{\frac {3}{x^2}}+\frac {3}{x^2}}}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3}+\frac {e^{2 e^{\frac {3}{x^2}}} x^2 \left (-36+e^{e^{\frac {3}{x^2}}} x\right )}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3}\right ) \, dx \\ & = 144 \int \frac {e^{2 e^{\frac {3}{x^2}}+\frac {3}{x^2}}}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx+\int \frac {e^{2 e^{\frac {3}{x^2}}} x^2 \left (-36+e^{e^{\frac {3}{x^2}}} x\right )}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx \\ & = 144 \int \frac {e^{2 e^{\frac {3}{x^2}}+\frac {3}{x^2}}}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx+\int \left (-\frac {24 e^{2 e^{\frac {3}{x^2}}} x^2}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3}+\frac {e^{2 e^{\frac {3}{x^2}}} x^2}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^2}\right ) \, dx \\ & = -\left (24 \int \frac {e^{2 e^{\frac {3}{x^2}}} x^2}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx\right )+144 \int \frac {e^{2 e^{\frac {3}{x^2}}+\frac {3}{x^2}}}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^3} \, dx+\int \frac {e^{2 e^{\frac {3}{x^2}}} x^2}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {e^{2 e^{\frac {3}{x^2}}} x^3}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^2} \]

[In]

Integrate[(E^(3*E^(3/x^2))*x^3 + E^(2*E^(3/x^2))*(144*E^(3/x^2) - 36*x^2))/(-1728 + 432*E^E^(3/x^2)*x - 36*E^(
2*E^(3/x^2))*x^2 + E^(3*E^(3/x^2))*x^3),x]

[Out]

(E^(2*E^(3/x^2))*x^3)/(-12 + E^E^(3/x^2)*x)^2

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {x^{3} {\mathrm e}^{2 \,{\mathrm e}^{\frac {3}{x^{2}}}}}{\left (x \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x^{2}}}}-12\right )^{2}}\)	\(27\)
parallelrisch	\(\frac {{\mathrm e}^{2 \,{\mathrm e}^{\frac {3}{x^{2}}}} x^{3}}{x^{2} {\mathrm e}^{2 \,{\mathrm e}^{\frac {3}{x^{2}}}}-24 x \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x^{2}}}}+144}\)	\(41\)

[In]

int((x^3*exp(exp(3/x^2))^3+(144*exp(3/x^2)-36*x^2)*exp(exp(3/x^2))^2)/(x^3*exp(exp(3/x^2))^3-36*x^2*exp(exp(3/
x^2))^2+432*x*exp(exp(3/x^2))-1728),x,method=_RETURNVERBOSE)

[Out]

x^3*exp(2*exp(3/x^2))/(x*exp(exp(3/x^2))-12)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {x^{3} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )}}{x^{2} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )} - 24 \, x e^{\left (e^{\left (\frac {3}{x^{2}}\right )}\right )} + 144} \]

[In]

integrate((x^3*exp(exp(3/x^2))^3+(144*exp(3/x^2)-36*x^2)*exp(exp(3/x^2))^2)/(x^3*exp(exp(3/x^2))^3-36*x^2*exp(
exp(3/x^2))^2+432*x*exp(exp(3/x^2))-1728),x, algorithm="fricas")

[Out]

x^3*e^(2*e^(3/x^2))/(x^2*e^(2*e^(3/x^2)) - 24*x*e^(e^(3/x^2)) + 144)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=x + \frac {24 x^{2} e^{e^{\frac {3}{x^{2}}}} - 144 x}{x^{2} e^{2 e^{\frac {3}{x^{2}}}} - 24 x e^{e^{\frac {3}{x^{2}}}} + 144} \]

[In]

integrate((x**3*exp(exp(3/x**2))**3+(144*exp(3/x**2)-36*x**2)*exp(exp(3/x**2))**2)/(x**3*exp(exp(3/x**2))**3-3
6*x**2*exp(exp(3/x**2))**2+432*x*exp(exp(3/x**2))-1728),x)

[Out]

x + (24*x**2*exp(exp(3/x**2)) - 144*x)/(x**2*exp(2*exp(3/x**2)) - 24*x*exp(exp(3/x**2)) + 144)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {x^{3} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )}}{x^{2} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )} - 24 \, x e^{\left (e^{\left (\frac {3}{x^{2}}\right )}\right )} + 144} \]

[In]

integrate((x^3*exp(exp(3/x^2))^3+(144*exp(3/x^2)-36*x^2)*exp(exp(3/x^2))^2)/(x^3*exp(exp(3/x^2))^3-36*x^2*exp(
exp(3/x^2))^2+432*x*exp(exp(3/x^2))-1728),x, algorithm="maxima")

[Out]

x^3*e^(2*e^(3/x^2))/(x^2*e^(2*e^(3/x^2)) - 24*x*e^(e^(3/x^2)) + 144)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (19) = 38\).

Time = 0.30 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {x^{3} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )}}{x^{2} e^{\left (2 \, e^{\left (\frac {3}{x^{2}}\right )}\right )} - 24 \, x e^{\left (e^{\left (\frac {3}{x^{2}}\right )}\right )} + 144} \]

[In]

integrate((x^3*exp(exp(3/x^2))^3+(144*exp(3/x^2)-36*x^2)*exp(exp(3/x^2))^2)/(x^3*exp(exp(3/x^2))^3-36*x^2*exp(
exp(3/x^2))^2+432*x*exp(exp(3/x^2))-1728),x, algorithm="giac")

[Out]

x^3*e^(2*e^(3/x^2))/(x^2*e^(2*e^(3/x^2)) - 24*x*e^(e^(3/x^2)) + 144)

Mupad [B] (verification not implemented)

Time = 7.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{3 e^{\frac {3}{x^2}}} x^3+e^{2 e^{\frac {3}{x^2}}} \left (144 e^{\frac {3}{x^2}}-36 x^2\right )}{-1728+432 e^{e^{\frac {3}{x^2}}} x-36 e^{2 e^{\frac {3}{x^2}}} x^2+e^{3 e^{\frac {3}{x^2}}} x^3} \, dx=\frac {x^3\,{\mathrm {e}}^{2\,{\mathrm {e}}^{\frac {3}{x^2}}}}{{\left (x\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {3}{x^2}}}-12\right )}^2} \]

[In]

int(-(exp(2*exp(3/x^2))*(144*exp(3/x^2) - 36*x^2) + x^3*exp(3*exp(3/x^2)))/(36*x^2*exp(2*exp(3/x^2)) - x^3*exp
(3*exp(3/x^2)) - 432*x*exp(exp(3/x^2)) + 1728),x)

[Out]

(x^3*exp(2*exp(3/x^2)))/(x*exp(exp(3/x^2)) - 12)^2