3.81.67 \(\int \frac {e^{2 e^2-2 x} (4-12 x)+9 x^4+e^{e^2-x} (-12 x^2-9 x^4)}{4 e^{2 e^2-2 x}-12 e^{e^2-x} x^2+9 x^4} \, dx\)

Optimal. Leaf size=27 \[ 3+e^5-\frac {3}{-3 e^{-e^2+x}+\frac {2}{x^2}}+x \]

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Rubi [F]  time = 2.83, 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^{2 e^2-2 x} (4-12 x)+9 x^4+e^{e^2-x} \left (-12 x^2-9 x^4\right )}{4 e^{2 e^2-2 x}-12 e^{e^2-x} x^2+9 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*E^2 - 2*x)*(4 - 12*x) + 9*x^4 + E^(E^2 - x)*(-12*x^2 - 9*x^4))/(4*E^(2*E^2 - 2*x) - 12*E^(E^2 - x)*x
^2 + 9*x^4),x]

[Out]

x - (3*x^2)/2 - (9*E^(-E^2 + x)*x^4)/4 - 27*Defer[Int][(E^(2*x)*x^5)/(-2*E^E^2 + 3*E^x*x^2)^2, x] - (27*Defer[
Int][(E^(2*x)*x^6)/(-2*E^E^2 + 3*E^x*x^2)^2, x])/2 + 27*Defer[Int][(E^(-E^2 + 2*x)*x^5)/(-2*E^E^2 + 3*E^x*x^2)
, x] + (27*Defer[Int][(E^(-E^2 + 2*x)*x^6)/(-2*E^E^2 + 3*E^x*x^2), x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (e^{2 e^2-2 x} (4-12 x)+9 x^4+e^{e^2-x} \left (-12 x^2-9 x^4\right )\right )}{\left (2 e^{e^2}-3 e^x x^2\right )^2} \, dx\\ &=\int \left (1-3 x-\frac {9}{4} e^{-e^2+x} x^3 (4+x)-\frac {27 e^{-e^2+2 x} x^5 (4+x)}{4 \left (2 e^{e^2}-3 e^x x^2\right )}-\frac {27 e^{2 x} x^5 (2+x)}{2 \left (-2 e^{e^2}+3 e^x x^2\right )^2}\right ) \, dx\\ &=x-\frac {3 x^2}{2}-\frac {9}{4} \int e^{-e^2+x} x^3 (4+x) \, dx-\frac {27}{4} \int \frac {e^{-e^2+2 x} x^5 (4+x)}{2 e^{e^2}-3 e^x x^2} \, dx-\frac {27}{2} \int \frac {e^{2 x} x^5 (2+x)}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx\\ &=x-\frac {3 x^2}{2}-\frac {9}{4} \int \left (4 e^{-e^2+x} x^3+e^{-e^2+x} x^4\right ) \, dx-\frac {27}{4} \int \left (-\frac {4 e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2}-\frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2}\right ) \, dx-\frac {27}{2} \int \left (\frac {2 e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2}+\frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2}\right ) \, dx\\ &=x-\frac {3 x^2}{2}-\frac {9}{4} \int e^{-e^2+x} x^4 \, dx+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx-9 \int e^{-e^2+x} x^3 \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx\\ &=x-\frac {3 x^2}{2}-9 e^{-e^2+x} x^3-\frac {9}{4} e^{-e^2+x} x^4+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx+9 \int e^{-e^2+x} x^3 \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int e^{-e^2+x} x^2 \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx\\ &=x-\frac {3 x^2}{2}+27 e^{-e^2+x} x^2-\frac {9}{4} e^{-e^2+x} x^4+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx-27 \int e^{-e^2+x} x^2 \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx-54 \int e^{-e^2+x} x \, dx\\ &=x-54 e^{-e^2+x} x-\frac {3 x^2}{2}-\frac {9}{4} e^{-e^2+x} x^4+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx+54 \int e^{-e^2+x} \, dx+54 \int e^{-e^2+x} x \, dx\\ &=54 e^{-e^2+x}+x-\frac {3 x^2}{2}-\frac {9}{4} e^{-e^2+x} x^4+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx-54 \int e^{-e^2+x} \, dx\\ &=x-\frac {3 x^2}{2}-\frac {9}{4} e^{-e^2+x} x^4+\frac {27}{4} \int \frac {e^{-e^2+2 x} x^6}{-2 e^{e^2}+3 e^x x^2} \, dx-\frac {27}{2} \int \frac {e^{2 x} x^6}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx-27 \int \frac {e^{2 x} x^5}{\left (-2 e^{e^2}+3 e^x x^2\right )^2} \, dx+27 \int \frac {e^{-e^2+2 x} x^5}{-2 e^{e^2}+3 e^x x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 30, normalized size = 1.11 \begin {gather*} x+\frac {3 e^{e^2} x^2}{-2 e^{e^2}+3 e^x x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^2 - 2*x)*(4 - 12*x) + 9*x^4 + E^(E^2 - x)*(-12*x^2 - 9*x^4))/(4*E^(2*E^2 - 2*x) - 12*E^(E^2
- x)*x^2 + 9*x^4),x]

[Out]

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

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fricas [A]  time = 1.29, size = 41, normalized size = 1.52 \begin {gather*} \frac {3 \, x^{3} + {\left (3 \, x^{2} - 2 \, x\right )} e^{\left (-x + e^{2}\right )}}{3 \, x^{2} - 2 \, e^{\left (-x + e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+4)*exp(exp(1)^2-x)^2+(-9*x^4-12*x^2)*exp(exp(1)^2-x)+9*x^4)/(4*exp(exp(1)^2-x)^2-12*x^2*exp(
exp(1)^2-x)+9*x^4),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.26, size = 162, normalized size = 6.00 \begin {gather*} \frac {9 \, x^{6} e^{x} + 9 \, x^{5} e^{x} + 9 \, x^{5} e^{\left (e^{2}\right )} - 18 \, x^{4} e^{x} + 6 \, x^{4} e^{\left (e^{2}\right )} - 6 \, x^{3} e^{\left (-x + 2 \, e^{2}\right )} - 12 \, x^{3} e^{\left (e^{2}\right )} - 8 \, x^{2} e^{\left (-x + 2 \, e^{2}\right )} + 24 \, x^{2} e^{\left (e^{2}\right )} + 4 \, x e^{\left (-x + 2 \, e^{2}\right )} - 8 \, e^{\left (-x + 2 \, e^{2}\right )}}{9 \, x^{5} e^{x} + 18 \, x^{4} e^{x} - 12 \, x^{3} e^{\left (e^{2}\right )} - 24 \, x^{2} e^{\left (e^{2}\right )} + 4 \, x e^{\left (-x + 2 \, e^{2}\right )} + 8 \, e^{\left (-x + 2 \, e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+4)*exp(exp(1)^2-x)^2+(-9*x^4-12*x^2)*exp(exp(1)^2-x)+9*x^4)/(4*exp(exp(1)^2-x)^2-12*x^2*exp(
exp(1)^2-x)+9*x^4),x, algorithm="giac")

[Out]

(9*x^6*e^x + 9*x^5*e^x + 9*x^5*e^(e^2) - 18*x^4*e^x + 6*x^4*e^(e^2) - 6*x^3*e^(-x + 2*e^2) - 12*x^3*e^(e^2) -
8*x^2*e^(-x + 2*e^2) + 24*x^2*e^(e^2) + 4*x*e^(-x + 2*e^2) - 8*e^(-x + 2*e^2))/(9*x^5*e^x + 18*x^4*e^x - 12*x^
3*e^(e^2) - 24*x^2*e^(e^2) + 4*x*e^(-x + 2*e^2) + 8*e^(-x + 2*e^2))

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maple [A]  time = 0.31, size = 30, normalized size = 1.11

method	result	size
risch	\(-\frac {3 x^{2}}{2}+x +\frac {9 x^{4}}{2 \left (3 x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{2}-x}\right )}\)	\(30\)
norman	\(\frac {3 x^{3}-2 x \,{\mathrm e}^{{\mathrm e}^{2}-x}+3 x^{2} {\mathrm e}^{{\mathrm e}^{2}-x}}{3 x^{2}-2 \,{\mathrm e}^{{\mathrm e}^{2}-x}}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x+4)*exp(exp(1)^2-x)^2+(-9*x^4-12*x^2)*exp(exp(1)^2-x)+9*x^4)/(4*exp(exp(1)^2-x)^2-12*x^2*exp(exp(1)
^2-x)+9*x^4),x,method=_RETURNVERBOSE)

[Out]

-3/2*x^2+x+9/2*x^4/(3*x^2-2*exp(exp(2)-x))

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maxima [A]  time = 0.42, size = 38, normalized size = 1.41 \begin {gather*} \frac {3 \, x^{3} e^{x} + 3 \, x^{2} e^{\left (e^{2}\right )} - 2 \, x e^{\left (e^{2}\right )}}{3 \, x^{2} e^{x} - 2 \, e^{\left (e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+4)*exp(exp(1)^2-x)^2+(-9*x^4-12*x^2)*exp(exp(1)^2-x)+9*x^4)/(4*exp(exp(1)^2-x)^2-12*x^2*exp(
exp(1)^2-x)+9*x^4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.11, size = 29, normalized size = 1.07 \begin {gather*} x-\frac {9\,x^4}{2\,\left (2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\mathrm {e}}^2}-3\,x^2\right )}-\frac {3\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(2) - x)*(12*x^2 + 9*x^4) - 9*x^4 + exp(2*exp(2) - 2*x)*(12*x - 4))/(4*exp(2*exp(2) - 2*x) - 12*x
^2*exp(exp(2) - x) + 9*x^4),x)

[Out]

x - (9*x^4)/(2*(2*exp(-x)*exp(exp(2)) - 3*x^2)) - (3*x^2)/2

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sympy [A]  time = 0.14, size = 26, normalized size = 0.96 \begin {gather*} - \frac {9 x^{4}}{- 6 x^{2} + 4 e^{- x + e^{2}}} - \frac {3 x^{2}}{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x+4)*exp(exp(1)**2-x)**2+(-9*x**4-12*x**2)*exp(exp(1)**2-x)+9*x**4)/(4*exp(exp(1)**2-x)**2-12*
x**2*exp(exp(1)**2-x)+9*x**4),x)

[Out]

-9*x**4/(-6*x**2 + 4*exp(-x + exp(2))) - 3*x**2/2 + x

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