3.68.48 \(\int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} (e^{4+x} (-12 e^{16}+12 x)+e^{4+\frac {e^{16}}{x}+x} (4 x^2+4 x^3)))}{x^2} \, dx\)

Optimal. Leaf size=28 \[ \left (e^{4+x}+2 e^{-\frac {3 e^{-\frac {e^{16}}{x}}}{x}} x\right )^2 \]

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Rubi [F]  time = 7.32, 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^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-24 e^{16} x+24 x^2+2 e^{8+\frac {e^{16}}{x}+\frac {6 e^{-\frac {e^{16}}{x}}}{x}+2 x} x^2+8 e^{\frac {e^{16}}{x}} x^3+e^{\frac {3 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+x} \left (-12 e^{16}+12 x\right )+e^{4+\frac {e^{16}}{x}+x} \left (4 x^2+4 x^3\right )\right )\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-(E^16/x) - 6/(E^(E^16/x)*x))*(-24*E^16*x + 24*x^2 + 2*E^(8 + E^16/x + 6/(E^(E^16/x)*x) + 2*x)*x^2 + 8
*E^(E^16/x)*x^3 + E^(3/(E^(E^16/x)*x))*(E^(4 + x)*(-12*E^16 + 12*x) + E^(4 + E^16/x + x)*(4*x^2 + 4*x^3))))/x^
2,x]

[Out]

E^(8 + 2*x) + 24*Defer[Int][E^(-(E^16/x) - 6/(E^(E^16/x)*x)), x] + 4*Defer[Int][E^(4 - 3/(E^(E^16/x)*x) + x),
x] - 12*Defer[Int][E^(20 - E^16/x - 3/(E^(E^16/x)*x) + x)/x^2, x] + 12*Defer[Int][E^(4 - E^16/x - 3/(E^(E^16/x
)*x) + x)/x, x] + 8*Defer[Int][x/E^(6/(E^(E^16/x)*x)), x] + 4*Defer[Int][E^(4 - 3/(E^(E^16/x)*x) + x)*x, x] +
24*Defer[Subst][Defer[Int][E^(16 - E^16*x - (6*x)/E^(E^16*x))/x, x], x, x^(-1)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+2 x\right ) \left (-6 e^{16}+6 x+2 e^{\frac {e^{16}}{x}} x^2+e^{\frac {e^{16}+3 e^{-\frac {e^{16}}{x}}+x (4+x)}{x}} x^2\right )}{x^2} \, dx\\ &=2 \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+2 x\right ) \left (-6 e^{16}+6 x+2 e^{\frac {e^{16}}{x}} x^2+e^{\frac {e^{16}+3 e^{-\frac {e^{16}}{x}}+x (4+x)}{x}} x^2\right )}{x^2} \, dx\\ &=2 \int \left (e^{8+2 x}+\frac {4 e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-3 e^{16}+3 x+e^{\frac {e^{16}}{x}} x^2\right )}{x}-\frac {2 e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \left (3 e^{16}-3 x-e^{\frac {e^{16}}{x}} x^2-e^{\frac {e^{16}}{x}} x^3\right )}{x^2}\right ) \, dx\\ &=2 \int e^{8+2 x} \, dx-4 \int \frac {e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \left (3 e^{16}-3 x-e^{\frac {e^{16}}{x}} x^2-e^{\frac {e^{16}}{x}} x^3\right )}{x^2} \, dx+8 \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (-3 e^{16}+3 x+e^{\frac {e^{16}}{x}} x^2\right )}{x} \, dx\\ &=e^{8+2 x}-4 \int \frac {e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \left (3 e^{16}-3 x-e^{\frac {e^{16}}{x}} x^2 (1+x)\right )}{x^2} \, dx+8 \int \left (-\frac {3 e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{16}-x\right )}{x}+e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} x\right ) \, dx\\ &=e^{8+2 x}-4 \int \left (\frac {3 e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \left (e^{16}-x\right )}{x^2}-e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} (1+x)\right ) \, dx+8 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} x \, dx-24 \int \frac {e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{16}-x\right )}{x} \, dx\\ &=e^{8+2 x}+4 \int e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} (1+x) \, dx+8 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} x \, dx-12 \int \frac {e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \left (e^{16}-x\right )}{x^2} \, dx-24 \int \left (-e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}}+\frac {e^{16-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}}}{x}\right ) \, dx\\ &=e^{8+2 x}+4 \int \left (e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} x\right ) \, dx+8 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} x \, dx-12 \int \left (\frac {e^{20-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}}{x^2}-\frac {e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}}{x}\right ) \, dx+24 \int e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \, dx-24 \int \frac {e^{16-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}}}{x} \, dx\\ &=e^{8+2 x}+4 \int e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} \, dx+4 \int e^{4-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x} x \, dx+8 \int e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} x \, dx-12 \int \frac {e^{20-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}}{x^2} \, dx+12 \int \frac {e^{4-\frac {e^{16}}{x}-\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}}{x} \, dx+24 \int e^{-\frac {e^{16}}{x}-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \, dx+24 \operatorname {Subst}\left (\int \frac {e^{16-e^{16} x-6 e^{-e^{16} x} x}}{x} \, dx,x,\frac {1}{x}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 44, normalized size = 1.57 \begin {gather*} e^{-\frac {6 e^{-\frac {e^{16}}{x}}}{x}} \left (e^{4+\frac {3 e^{-\frac {e^{16}}{x}}}{x}+x}+2 x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-(E^16/x) - 6/(E^(E^16/x)*x))*(-24*E^16*x + 24*x^2 + 2*E^(8 + E^16/x + 6/(E^(E^16/x)*x) + 2*x)*x
^2 + 8*E^(E^16/x)*x^3 + E^(3/(E^(E^16/x)*x))*(E^(4 + x)*(-12*E^16 + 12*x) + E^(4 + E^16/x + x)*(4*x^2 + 4*x^3)
)))/x^2,x]

[Out]

(E^(4 + 3/(E^(E^16/x)*x) + x) + 2*x)^2/E^(6/(E^(E^16/x)*x))

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fricas [B]  time = 0.63, size = 103, normalized size = 3.68 \begin {gather*} {\left (4 \, x^{2} e^{\left (\frac {2 \, e^{16}}{x}\right )} + 4 \, x e^{\left (\frac {x^{2} + 4 \, x + e^{16}}{x} + \frac {e^{16}}{x} + \frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )} + e^{\left (\frac {2 \, {\left (x^{2} + 4 \, x + e^{16}\right )}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {2 \, e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-
12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x
)/exp(3/x/exp(exp(16)/x))^2,x, algorithm="fricas")

[Out]

(4*x^2*e^(2*e^16/x) + 4*x*e^((x^2 + 4*x + e^16)/x + e^16/x + 3*e^(-e^16/x)/x) + e^(2*(x^2 + 4*x + e^16)/x + 6*
e^(-e^16/x)/x))*e^(-2*e^16/x - 6*e^(-e^16/x)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (4 \, x^{3} e^{\left (\frac {e^{16}}{x}\right )} + x^{2} e^{\left (2 \, x + \frac {e^{16}}{x} + \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 8\right )} + 12 \, x^{2} - 12 \, x e^{16} + 2 \, {\left ({\left (x^{3} + x^{2}\right )} e^{\left (x + \frac {e^{16}}{x} + 4\right )} + 3 \, {\left (x - e^{16}\right )} e^{\left (x + 4\right )}\right )} e^{\left (\frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-
12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x
)/exp(3/x/exp(exp(16)/x))^2,x, algorithm="giac")

[Out]

integrate(2*(4*x^3*e^(e^16/x) + x^2*e^(2*x + e^16/x + 6*e^(-e^16/x)/x + 8) + 12*x^2 - 12*x*e^16 + 2*((x^3 + x^
2)*e^(x + e^16/x + 4) + 3*(x - e^16)*e^(x + 4))*e^(3*e^(-e^16/x)/x))*e^(-e^16/x - 6*e^(-e^16/x)/x)/x^2, x)

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maple [A]  time = 0.07, size = 52, normalized size = 1.86




method result size



risch \({\mathrm e}^{2 x +8}+4 x \,{\mathrm e}^{\frac {x^{2}-3 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}+4 x}{x}}+4 x^{2} {\mathrm e}^{-\frac {6 \,{\mathrm e}^{-\frac {{\mathrm e}^{16}}{x}}}{x}}\) \(52\)



Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-12*exp
(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x)/exp(
3/x/exp(exp(16)/x))^2,x,method=_RETURNVERBOSE)

[Out]

exp(2*x+8)+4*x*exp((x^2-3*exp(-exp(16)/x)+4*x)/x)+4*x^2*exp(-6/x*exp(-exp(16)/x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4 \, x e^{\left (x - \frac {3 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x} + 4\right )} + e^{\left (2 \, x + 8\right )} + 2 \, \int \frac {4 \, {\left (x^{2} e^{\left (\frac {e^{16}}{x}\right )} + 3 \, x - 3 \, e^{16}\right )} e^{\left (-\frac {e^{16}}{x} - \frac {6 \, e^{\left (-\frac {e^{16}}{x}\right )}}{x}\right )}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(4+x)^2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))^2+((4*x^3+4*x^2)*exp(4+x)*exp(exp(16)/x)+(-
12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(16)/x))+8*x^3*exp(exp(16)/x)-24*x*exp(16)+24*x^2)/x^2/exp(exp(16)/x
)/exp(3/x/exp(exp(16)/x))^2,x, algorithm="maxima")

[Out]

4*x*e^(x - 3*e^(-e^16/x)/x + 4) + e^(2*x + 8) + 2*integrate(4*(x^2*e^(e^16/x) + 3*x - 3*e^16)*e^(-e^16/x - 6*e
^(-e^16/x)/x)/x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left (8\,x^3\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}-24\,x\,{\mathrm {e}}^{16}+{\mathrm {e}}^{\frac {3\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\,\left ({\mathrm {e}}^{x+4}\,\left (12\,x-12\,{\mathrm {e}}^{16}\right )+{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{x+4}\,\left (4\,x^3+4\,x^2\right )\right )+24\,x^2+2\,x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{16}}{x}}\,{\mathrm {e}}^{2\,x+8}\,{\mathrm {e}}^{\frac {6\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{16}}{x}}}{x}}\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-exp(16)/x)*exp(-(6*exp(-exp(16)/x))/x)*(8*x^3*exp(exp(16)/x) - 24*x*exp(16) + exp((3*exp(-exp(16)/x)
)/x)*(exp(x + 4)*(12*x - 12*exp(16)) + exp(exp(16)/x)*exp(x + 4)*(4*x^2 + 4*x^3)) + 24*x^2 + 2*x^2*exp(exp(16)
/x)*exp(2*x + 8)*exp((6*exp(-exp(16)/x))/x)))/x^2,x)

[Out]

int((exp(-exp(16)/x)*exp(-(6*exp(-exp(16)/x))/x)*(8*x^3*exp(exp(16)/x) - 24*x*exp(16) + exp((3*exp(-exp(16)/x)
)/x)*(exp(x + 4)*(12*x - 12*exp(16)) + exp(exp(16)/x)*exp(x + 4)*(4*x^2 + 4*x^3)) + 24*x^2 + 2*x^2*exp(exp(16)
/x)*exp(2*x + 8)*exp((6*exp(-exp(16)/x))/x)))/x^2, x)

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sympy [F(-1)]  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((2*x**2*exp(4+x)**2*exp(exp(16)/x)*exp(3/x/exp(exp(16)/x))**2+((4*x**3+4*x**2)*exp(4+x)*exp(exp(16)/
x)+(-12*exp(16)+12*x)*exp(4+x))*exp(3/x/exp(exp(16)/x))+8*x**3*exp(exp(16)/x)-24*x*exp(16)+24*x**2)/x**2/exp(e
xp(16)/x)/exp(3/x/exp(exp(16)/x))**2,x)

[Out]

Timed out

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