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

Optimal. Leaf size=23 \[ \frac {x^3}{\left (12 e^{-e^{\frac {3}{x^2}}}-x\right )^2} \]

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Rubi [F]  time = 2.00, 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*} \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 \end {gather*}

Verification is not applicable to the result.

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

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Mathematica [A]  time = 0.38, size = 30, normalized size = 1.30 \begin {gather*} \frac {e^{2 e^{\frac {3}{x^2}}} x^3}{\left (-12+e^{e^{\frac {3}{x^2}}} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.58, size = 40, normalized size = 1.74 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.56, size = 40, normalized size = 1.74 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.07, size = 30, normalized size = 1.30

method	result	size
risch	\(x +\frac {24 \left (x \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x^{2}}}}-6\right ) x}{\left (x \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x^{2}}}}-12\right )^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[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+24*(x*exp(exp(3/x^2))-6)*x/(x*exp(exp(3/x^2))-12)^2

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maxima [B]  time = 0.42, size = 40, normalized size = 1.74 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.53, size = 26, normalized size = 1.13 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [B]  time = 0.31, size = 44, normalized size = 1.91 \begin {gather*} 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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