3.57.45 \(\int \frac {6 e^{3 e^{4/x}} x+18 e^{2 e^{4/x}} x^2+6 x^4+e^{e^{4/x}} (-24 e^{4/x}-6 x+18 x^3)}{e^{3 e^{4/x}}+3 e^{2 e^{4/x}} x+3 e^{e^{4/x}} x^2+x^3} \, dx\)

Optimal. Leaf size=25 \[ 3 \left (4+x^2-\frac {x^2}{\left (e^{e^{4/x}}+x\right )^2}\right ) \]

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Rubi [F]  time = 1.44, 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 {6 e^{3 e^{4/x}} x+18 e^{2 e^{4/x}} x^2+6 x^4+e^{e^{4/x}} \left (-24 e^{4/x}-6 x+18 x^3\right )}{e^{3 e^{4/x}}+3 e^{2 e^{4/x}} x+3 e^{e^{4/x}} x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6*E^(3*E^(4/x))*x + 18*E^(2*E^(4/x))*x^2 + 6*x^4 + E^E^(4/x)*(-24*E^(4/x) - 6*x + 18*x^3))/(E^(3*E^(4/x))
+ 3*E^(2*E^(4/x))*x + 3*E^E^(4/x)*x^2 + x^3),x]

[Out]

-24*Defer[Int][E^(E^(4/x) + 4/x)/(E^E^(4/x) + x)^3, x] - 6*Defer[Int][(E^E^(4/x)*x)/(E^E^(4/x) + x)^3, x] + 6*
Defer[Int][(E^(3*E^(4/x))*x)/(E^E^(4/x) + x)^3, x] + 18*Defer[Int][(E^(2*E^(4/x))*x^2)/(E^E^(4/x) + x)^3, x] +
18*Defer[Int][(E^E^(4/x)*x^3)/(E^E^(4/x) + x)^3, x] + 6*Defer[Int][x^4/(E^E^(4/x) + x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6 e^{3 e^{4/x}} x+18 e^{2 e^{4/x}} x^2+6 x^4+e^{e^{4/x}} \left (-24 e^{4/x}-6 x+18 x^3\right )}{\left (e^{e^{4/x}}+x\right )^3} \, dx\\ &=\int \left (-\frac {24 e^{e^{4/x}+\frac {4}{x}}}{\left (e^{e^{4/x}}+x\right )^3}-\frac {6 e^{e^{4/x}} x}{\left (e^{e^{4/x}}+x\right )^3}+\frac {6 e^{3 e^{4/x}} x}{\left (e^{e^{4/x}}+x\right )^3}+\frac {18 e^{2 e^{4/x}} x^2}{\left (e^{e^{4/x}}+x\right )^3}+\frac {18 e^{e^{4/x}} x^3}{\left (e^{e^{4/x}}+x\right )^3}+\frac {6 x^4}{\left (e^{e^{4/x}}+x\right )^3}\right ) \, dx\\ &=-\left (6 \int \frac {e^{e^{4/x}} x}{\left (e^{e^{4/x}}+x\right )^3} \, dx\right )+6 \int \frac {e^{3 e^{4/x}} x}{\left (e^{e^{4/x}}+x\right )^3} \, dx+6 \int \frac {x^4}{\left (e^{e^{4/x}}+x\right )^3} \, dx+18 \int \frac {e^{2 e^{4/x}} x^2}{\left (e^{e^{4/x}}+x\right )^3} \, dx+18 \int \frac {e^{e^{4/x}} x^3}{\left (e^{e^{4/x}}+x\right )^3} \, dx-24 \int \frac {e^{e^{4/x}+\frac {4}{x}}}{\left (e^{e^{4/x}}+x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 22, normalized size = 0.88 \begin {gather*} 3 x^2 \left (1-\frac {1}{\left (e^{e^{4/x}}+x\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*E^(3*E^(4/x))*x + 18*E^(2*E^(4/x))*x^2 + 6*x^4 + E^E^(4/x)*(-24*E^(4/x) - 6*x + 18*x^3))/(E^(3*E^
(4/x)) + 3*E^(2*E^(4/x))*x + 3*E^E^(4/x)*x^2 + x^3),x]

[Out]

3*x^2*(1 - (E^E^(4/x) + x)^(-2))

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fricas [B]  time = 0.65, size = 61, normalized size = 2.44 \begin {gather*} \frac {3 \, {\left (x^{4} + 2 \, x^{3} e^{\left (e^{\frac {4}{x}}\right )} + x^{2} e^{\left (2 \, e^{\frac {4}{x}}\right )} - x^{2}\right )}}{x^{2} + 2 \, x e^{\left (e^{\frac {4}{x}}\right )} + e^{\left (2 \, e^{\frac {4}{x}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(exp(4/x))^3+18*x^2*exp(exp(4/x))^2+(-24*exp(4/x)+18*x^3-6*x)*exp(exp(4/x))+6*x^4)/(exp(exp(
4/x))^3+3*x*exp(exp(4/x))^2+3*x^2*exp(exp(4/x))+x^3),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.18, size = 61, normalized size = 2.44 \begin {gather*} \frac {3 \, {\left (x^{4} + 2 \, x^{3} e^{\left (e^{\frac {4}{x}}\right )} + x^{2} e^{\left (2 \, e^{\frac {4}{x}}\right )} - x^{2}\right )}}{x^{2} + 2 \, x e^{\left (e^{\frac {4}{x}}\right )} + e^{\left (2 \, e^{\frac {4}{x}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(exp(4/x))^3+18*x^2*exp(exp(4/x))^2+(-24*exp(4/x)+18*x^3-6*x)*exp(exp(4/x))+6*x^4)/(exp(exp(
4/x))^3+3*x*exp(exp(4/x))^2+3*x^2*exp(exp(4/x))+x^3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 23, normalized size = 0.92

method	result	size
risch	\(3 x^{2}-\frac {3 x^{2}}{\left ({\mathrm e}^{{\mathrm e}^{\frac {4}{x}}}+x \right )^{2}}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*exp(exp(4/x))^3+18*x^2*exp(exp(4/x))^2+(-24*exp(4/x)+18*x^3-6*x)*exp(exp(4/x))+6*x^4)/(exp(exp(4/x))^
3+3*x*exp(exp(4/x))^2+3*x^2*exp(exp(4/x))+x^3),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.41, size = 61, normalized size = 2.44 \begin {gather*} \frac {3 \, {\left (x^{4} + 2 \, x^{3} e^{\left (e^{\frac {4}{x}}\right )} + x^{2} e^{\left (2 \, e^{\frac {4}{x}}\right )} - x^{2}\right )}}{x^{2} + 2 \, x e^{\left (e^{\frac {4}{x}}\right )} + e^{\left (2 \, e^{\frac {4}{x}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(exp(4/x))^3+18*x^2*exp(exp(4/x))^2+(-24*exp(4/x)+18*x^3-6*x)*exp(exp(4/x))+6*x^4)/(exp(exp(
4/x))^3+3*x*exp(exp(4/x))^2+3*x^2*exp(exp(4/x))+x^3),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.92, size = 44, normalized size = 1.76 \begin {gather*} \frac {6\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{4/x}}+3\,x^2\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^{4/x}}+x^2-1\right )}{{\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4/x}}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*exp(3*exp(4/x)) + 18*x^2*exp(2*exp(4/x)) - exp(exp(4/x))*(6*x + 24*exp(4/x) - 18*x^3) + 6*x^4)/(exp(3
*exp(4/x)) + 3*x*exp(2*exp(4/x)) + 3*x^2*exp(exp(4/x)) + x^3),x)

[Out]

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

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sympy [A]  time = 0.39, size = 31, normalized size = 1.24 \begin {gather*} 3 x^{2} - \frac {3 x^{2}}{x^{2} + 2 x e^{e^{\frac {4}{x}}} + e^{2 e^{\frac {4}{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(exp(4/x))**3+18*x**2*exp(exp(4/x))**2+(-24*exp(4/x)+18*x**3-6*x)*exp(exp(4/x))+6*x**4)/(exp
(exp(4/x))**3+3*x*exp(exp(4/x))**2+3*x**2*exp(exp(4/x))+x**3),x)

[Out]

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

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