3.87.79 \(\int \frac {e^{4 e^{e^{3/x}}} (10 e^{2/x}-5 x^2+e^{e^{3/x}+\frac {3}{x}} (-60+60 e^{2/x}+60 x))}{36 x^2+e^{4 e^{e^{3/x}}} (-12 x^2+12 e^{2/x} x^2+12 x^3)+e^{8 e^{e^{3/x}}} (x^2+e^{4/x} x^2-2 x^3+x^4+e^{2/x} (-2 x^2+2 x^3))} \, dx\) [8679]

Optimal. Leaf size=30 \[ \frac {5}{6+e^{4 e^{e^{3/x}}} \left (-1+e^{2/x}+x\right )} \]

[Out]

5/(exp(4*exp(exp(3/x)))*(x+exp(2/x)-1)+6)

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Rubi [F]
time = 36.19, 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^{4 e^{e^{3/x}}} \left (10 e^{2/x}-5 x^2+e^{e^{3/x}+\frac {3}{x}} \left (-60+60 e^{2/x}+60 x\right )\right )}{36 x^2+e^{4 e^{e^{3/x}}} \left (-12 x^2+12 e^{2/x} x^2+12 x^3\right )+e^{8 e^{e^{3/x}}} \left (x^2+e^{4/x} x^2-2 x^3+x^4+e^{2/x} \left (-2 x^2+2 x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4*E^E^(3/x))*(10*E^(2/x) - 5*x^2 + E^(E^(3/x) + 3/x)*(-60 + 60*E^(2/x) + 60*x)))/(36*x^2 + E^(4*E^E^(3
/x))*(-12*x^2 + 12*E^(2/x)*x^2 + 12*x^3) + E^(8*E^E^(3/x))*(x^2 + E^(4/x)*x^2 - 2*x^3 + x^4 + E^(2/x)*(-2*x^2
+ 2*x^3))),x]

[Out]

-5*Defer[Int][E^(4*E^E^(3/x))/(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2, x] - 60*Def
er[Int][1/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2), x] + 10*Defer[Int][E^(4*E
^E^(3/x))/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2), x] - 360*Defer[Int][E^(E^
(3/x) + x^(-1))/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2), x] + 2160*Defer[Int
][E^(-4*E^E^(3/x) + E^(3/x) + x^(-1))/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2
), x] - 10*Defer[Int][E^(4*E^E^(3/x))/(x*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)^2),
x] + 360*Defer[Int][E^(E^(3/x) + x^(-1))/(x*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)
^2), x] + 10*Defer[Int][1/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)), x] + 60*Def
er[Int][E^(E^(3/x) + x^(-1))/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*x)), x] - 720
*Defer[Int][E^(-4*E^E^(3/x) + E^(3/x) + x^(-1))/(x^2*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(
3/x))*x)), x] - 60*Defer[Int][E^(E^(3/x) + x^(-1))/(x*(6 - E^(4*E^E^(3/x)) + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^
(3/x))*x)), x] - 60*Defer[Subst][Defer[Int][E^(-4*E^x^3 + x^3), x], x, E^x^(-1)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^{4 e^{e^{3/x}}} \left (12 e^{e^{3/x}+\frac {5}{x}}+2 e^{2/x}+12 e^{e^{3/x}+\frac {3}{x}} (-1+x)-x^2\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\\ &=5 \int \frac {e^{4 e^{e^{3/x}}} \left (12 e^{e^{3/x}+\frac {5}{x}}+2 e^{2/x}+12 e^{e^{3/x}+\frac {3}{x}} (-1+x)-x^2\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\\ &=5 \int \left (\frac {12 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2}-\frac {2 e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}-6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}-\frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-2 e^{8 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}+72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+2 e^{8 e^{e^{3/x}}} x-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x+e^{8 e^{e^{3/x}}} x^2\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-2 e^{8 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}+72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+2 e^{8 e^{e^{3/x}}} x-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x+e^{8 e^{e^{3/x}}} x^2\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\right )-10 \int \frac {e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}-6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx+60 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2} \, dx\\ &=-\left (5 \int \frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} (-1+x)+e^{8 e^{e^{3/x}}} \left (-2+2 x+x^2\right )\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\right )-10 \int \frac {e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} (-1+x)\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right ) x^2} \, dx-60 \text {Subst}\left (\int e^{-4 e^{e^{3 x}}+e^{3 x}+x} \, dx,x,\frac {1}{x}\right )\\ &=-\left (5 \int \left (\frac {e^{4 e^{e^{3/x}}}}{\left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {12}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {2 e^{4 e^{e^{3/x}}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {72 e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {432 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {2 e^{4 e^{e^{3/x}}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {72 e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}\right ) \, dx\right )-10 \int \left (-\frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}-\frac {6 e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}+\frac {72 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}+\frac {6 e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}\right ) \, dx-60 \text {Subst}\left (\int e^{-4 e^{x^3}+x^3} \, dx,x,e^{\frac {1}{x}}\right )\\ &=-\left (5 \int \frac {e^{4 e^{e^{3/x}}}}{\left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\right )+10 \int \frac {e^{4 e^{e^{3/x}}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx-10 \int \frac {e^{4 e^{e^{3/x}}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+10 \int \frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \int \frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+60 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \text {Subst}\left (\int e^{-4 e^{x^3}+x^3} \, dx,x,e^{\frac {1}{x}}\right )-360 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+360 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx-720 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx+2160 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.15, size = 42, normalized size = 1.40 \begin {gather*} \frac {5}{6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*E^E^(3/x))*(10*E^(2/x) - 5*x^2 + E^(E^(3/x) + 3/x)*(-60 + 60*E^(2/x) + 60*x)))/(36*x^2 + E^(4*
E^E^(3/x))*(-12*x^2 + 12*E^(2/x)*x^2 + 12*x^3) + E^(8*E^E^(3/x))*(x^2 + E^(4/x)*x^2 - 2*x^3 + x^4 + E^(2/x)*(-
2*x^2 + 2*x^3))),x]

[Out]

5/(6 + E^(4*E^E^(3/x) + 2/x) + E^(4*E^E^(3/x))*(-1 + x))

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Maple [A]
time = 0.49, size = 49, normalized size = 1.63

method	result	size
risch	\(\frac {5}{{\mathrm e}^{\frac {4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}} x +2}{x}}+{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}}} x -{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}}}+6}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*exp(2/x)+60*x-60)*exp(3/x)*exp(exp(3/x))+10*exp(2/x)-5*x^2)*exp(4*exp(exp(3/x)))/((x^2*exp(2/x)^2+(2*
x^3-2*x^2)*exp(2/x)+x^4-2*x^3+x^2)*exp(4*exp(exp(3/x)))^2+(12*x^2*exp(2/x)+12*x^3-12*x^2)*exp(4*exp(exp(3/x)))
+36*x^2),x,method=_RETURNVERBOSE)

[Out]

5/(exp(2*(2*exp(exp(3/x))*x+1)/x)+exp(4*exp(exp(3/x)))*x-exp(4*exp(exp(3/x)))+6)

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Maxima [A]
time = 0.32, size = 26, normalized size = 0.87 \begin {gather*} \frac {5}{{\left (x + e^{\frac {2}{x}} - 1\right )} e^{\left (4 \, e^{\left (e^{\frac {3}{x}}\right )}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*exp(2/x)+60*x-60)*exp(3/x)*exp(exp(3/x))+10*exp(2/x)-5*x^2)*exp(4*exp(exp(3/x)))/((x^2*exp(2/x)
^2+(2*x^3-2*x^2)*exp(2/x)+x^4-2*x^3+x^2)*exp(4*exp(exp(3/x)))^2+(12*x^2*exp(2/x)+12*x^3-12*x^2)*exp(4*exp(exp(
3/x)))+36*x^2),x, algorithm="maxima")

[Out]

5/((x + e^(2/x) - 1)*e^(4*e^(e^(3/x))) + 6)

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Fricas [A]
time = 0.39, size = 40, normalized size = 1.33 \begin {gather*} \frac {5}{{\left (x + e^{\frac {2}{x}} - 1\right )} e^{\left (4 \, e^{\left (\frac {x e^{\frac {3}{x}} + 3}{x} - \frac {3}{x}\right )}\right )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*exp(2/x)+60*x-60)*exp(3/x)*exp(exp(3/x))+10*exp(2/x)-5*x^2)*exp(4*exp(exp(3/x)))/((x^2*exp(2/x)
^2+(2*x^3-2*x^2)*exp(2/x)+x^4-2*x^3+x^2)*exp(4*exp(exp(3/x)))^2+(12*x^2*exp(2/x)+12*x^3-12*x^2)*exp(4*exp(exp(
3/x)))+36*x^2),x, algorithm="fricas")

[Out]

5/((x + e^(2/x) - 1)*e^(4*e^((x*e^(3/x) + 3)/x - 3/x)) + 6)

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Sympy [A]
time = 0.32, size = 20, normalized size = 0.67 \begin {gather*} \frac {5}{\left (x + e^{\frac {2}{x}} - 1\right ) e^{4 e^{e^{\frac {3}{x}}}} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*exp(2/x)+60*x-60)*exp(3/x)*exp(exp(3/x))+10*exp(2/x)-5*x**2)*exp(4*exp(exp(3/x)))/((x**2*exp(2/
x)**2+(2*x**3-2*x**2)*exp(2/x)+x**4-2*x**3+x**2)*exp(4*exp(exp(3/x)))**2+(12*x**2*exp(2/x)+12*x**3-12*x**2)*ex
p(4*exp(exp(3/x)))+36*x**2),x)

[Out]

5/((x + exp(2/x) - 1)*exp(4*exp(exp(3/x))) + 6)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*exp(2/x)+60*x-60)*exp(3/x)*exp(exp(3/x))+10*exp(2/x)-5*x^2)*exp(4*exp(exp(3/x)))/((x^2*exp(2/x)
^2+(2*x^3-2*x^2)*exp(2/x)+x^4-2*x^3+x^2)*exp(4*exp(exp(3/x)))^2+(12*x^2*exp(2/x)+12*x^3-12*x^2)*exp(4*exp(exp(
3/x)))+36*x^2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (10\,{\mathrm {e}}^{2/x}-5\,x^2+{\mathrm {e}}^{{\mathrm {e}}^{3/x}}\,{\mathrm {e}}^{3/x}\,\left (60\,x+60\,{\mathrm {e}}^{2/x}-60\right )\right )}{{\mathrm {e}}^{8\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (x^2\,{\mathrm {e}}^{4/x}-{\mathrm {e}}^{2/x}\,\left (2\,x^2-2\,x^3\right )+x^2-2\,x^3+x^4\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (12\,x^2\,{\mathrm {e}}^{2/x}-12\,x^2+12\,x^3\right )+36\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*exp(exp(3/x)))*(10*exp(2/x) - 5*x^2 + exp(exp(3/x))*exp(3/x)*(60*x + 60*exp(2/x) - 60)))/(exp(8*exp
(exp(3/x)))*(x^2*exp(4/x) - exp(2/x)*(2*x^2 - 2*x^3) + x^2 - 2*x^3 + x^4) + exp(4*exp(exp(3/x)))*(12*x^2*exp(2
/x) - 12*x^2 + 12*x^3) + 36*x^2),x)

[Out]

int((exp(4*exp(exp(3/x)))*(10*exp(2/x) - 5*x^2 + exp(exp(3/x))*exp(3/x)*(60*x + 60*exp(2/x) - 60)))/(exp(8*exp
(exp(3/x)))*(x^2*exp(4/x) - exp(2/x)*(2*x^2 - 2*x^3) + x^2 - 2*x^3 + x^4) + exp(4*exp(exp(3/x)))*(12*x^2*exp(2
/x) - 12*x^2 + 12*x^3) + 36*x^2), x)

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