3.66.13 \(\int \frac {e^{\frac {x-e^{2 e^{2 x}} x+16 x^2-x^3+4 x^2 \log (4)}{4+x}} (4+128 x+4 x^2-2 x^3+e^{2 e^{2 x}} (-4+e^{2 x} (-16 x-4 x^2))+(32 x+4 x^2) \log (4))}{16+8 x+x^2} \, dx\)

Optimal. Leaf size=34 \[ e^{x \left (-x+\frac {1-e^{2 e^{2 x}}+4 x (5+\log (4))}{4+x}\right )} \]

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Rubi [F]  time = 36.52, 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 {\exp \left (\frac {x-e^{2 e^{2 x}} x+16 x^2-x^3+4 x^2 \log (4)}{4+x}\right ) \left (4+128 x+4 x^2-2 x^3+e^{2 e^{2 x}} \left (-4+e^{2 x} \left (-16 x-4 x^2\right )\right )+\left (32 x+4 x^2\right ) \log (4)\right )}{16+8 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((x - E^(2*E^(2*x))*x + 16*x^2 - x^3 + 4*x^2*Log[4])/(4 + x))*(4 + 128*x + 4*x^2 - 2*x^3 + E^(2*E^(2*x)
)*(-4 + E^(2*x)*(-16*x - 4*x^2)) + (32*x + 4*x^2)*Log[4]))/(16 + 8*x + x^2),x]

[Out]

-4*Defer[Int][E^(2*(E^(2*x) + x) + (x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x)), x] + 20*Defer
[Int][E^((x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x)), x] + 4*Log[4]*Defer[Int][E^((x - E^(2*E
^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x)), x] - 2*Defer[Int][E^((x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1
+ Log[2]/2))/(4 + x))*x, x] - 4*Defer[Int][E^(2*E^(2*x) + (x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))
/(4 + x))/(4 + x)^2, x] - 316*Defer[Int][E^((x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x))/(4 +
x)^2, x] - 64*Log[4]*Defer[Int][E^((x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x))/(4 + x)^2, x]
+ 16*Defer[Int][E^(2*(E^(2*x) + x) + (x - E^(2*E^(2*x))*x - x^3 + 16*x^2*(1 + Log[2]/2))/(4 + x))/(4 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x+16 x^2-x^3+4 x^2 \log (4)}{4+x}\right ) \left (4+128 x+4 x^2-2 x^3+e^{2 e^{2 x}} \left (-4+e^{2 x} \left (-16 x-4 x^2\right )\right )+\left (32 x+4 x^2\right ) \log (4)\right )}{(4+x)^2} \, dx\\ &=\int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) \left (4+128 x+4 x^2-2 x^3+e^{2 e^{2 x}} \left (-4+e^{2 x} \left (-16 x-4 x^2\right )\right )+\left (32 x+4 x^2\right ) \log (4)\right )}{(4+x)^2} \, dx\\ &=\int \left (\frac {4 \exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x}{-4-x}-\frac {4 \exp \left (2 e^{2 x}+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}+\frac {4 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}+\frac {128 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x}{(4+x)^2}+\frac {4 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x^2}{(4+x)^2}-\frac {2 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x^3}{(4+x)^2}+\frac {4 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x (8+x) \log (4)}{(4+x)^2}\right ) \, dx\\ &=-\left (2 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x^3}{(4+x)^2} \, dx\right )+4 \int \frac {\exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x}{-4-x} \, dx-4 \int \frac {\exp \left (2 e^{2 x}+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+4 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+4 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x^2}{(4+x)^2} \, dx+128 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x}{(4+x)^2} \, dx+(4 \log (4)) \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x (8+x)}{(4+x)^2} \, dx\\ &=-\left (2 \int \left (-8 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )+\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x-\frac {64 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}+\frac {48 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x}\right ) \, dx\right )-4 \int \frac {\exp \left (2 e^{2 x}+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+4 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+4 \int \left (-\exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )+\frac {4 \exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x}\right ) \, dx+4 \int \left (\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )+\frac {16 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}-\frac {8 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x}\right ) \, dx+128 \int \left (-\frac {4 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}+\frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x}\right ) \, dx+(4 \log (4)) \int \left (\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )-\frac {16 \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2}\right ) \, dx\\ &=-\left (2 \int \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) x \, dx\right )-4 \int \exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) \, dx+4 \int \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) \, dx-4 \int \frac {\exp \left (2 e^{2 x}+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+4 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+16 \int \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) \, dx+16 \int \frac {\exp \left (2 \left (e^{2 x}+x\right )+\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x} \, dx-32 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x} \, dx+64 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx-96 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x} \, dx+128 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+128 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{4+x} \, dx-512 \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx+(4 \log (4)) \int \exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right ) \, dx-(64 \log (4)) \int \frac {\exp \left (\frac {x-e^{2 e^{2 x}} x-x^3+16 x^2 \left (1+\frac {\log (2)}{2}\right )}{4+x}\right )}{(4+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 4.52, size = 40, normalized size = 1.18 \begin {gather*} 2^{\frac {8 x^2}{4+x}} e^{-\frac {x \left (-1+e^{2 e^{2 x}}-16 x+x^2\right )}{4+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((x - E^(2*E^(2*x))*x + 16*x^2 - x^3 + 4*x^2*Log[4])/(4 + x))*(4 + 128*x + 4*x^2 - 2*x^3 + E^(2*E
^(2*x))*(-4 + E^(2*x)*(-16*x - 4*x^2)) + (32*x + 4*x^2)*Log[4]))/(16 + 8*x + x^2),x]

[Out]

2^((8*x^2)/(4 + x))/E^((x*(-1 + E^(2*E^(2*x)) - 16*x + x^2))/(4 + x))

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fricas [A]  time = 1.08, size = 36, normalized size = 1.06 \begin {gather*} e^{\left (-\frac {x^{3} - 8 \, x^{2} \log \relax (2) - 16 \, x^{2} + x e^{\left (2 \, e^{\left (2 \, x\right )}\right )} - x}{x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x)^2-4)*exp(exp(x)^2)^2+2*(4*x^2+32*x)*log(2)-2*x^3+4*x^2+128*x+4)*exp((-x*exp(e
xp(x)^2)^2+8*x^2*log(2)-x^3+16*x^2+x)/(4+x))/(x^2+8*x+16),x, algorithm="fricas")

[Out]

e^(-(x^3 - 8*x^2*log(2) - 16*x^2 + x*e^(2*e^(2*x)) - x)/(x + 4))

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giac [A]  time = 0.35, size = 56, normalized size = 1.65 \begin {gather*} e^{\left (-\frac {x^{3}}{x + 4} + \frac {8 \, x^{2} \log \relax (2)}{x + 4} + \frac {16 \, x^{2}}{x + 4} - \frac {x e^{\left (2 \, e^{\left (2 \, x\right )}\right )}}{x + 4} + \frac {x}{x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x)^2-4)*exp(exp(x)^2)^2+2*(4*x^2+32*x)*log(2)-2*x^3+4*x^2+128*x+4)*exp((-x*exp(e
xp(x)^2)^2+8*x^2*log(2)-x^3+16*x^2+x)/(4+x))/(x^2+8*x+16),x, algorithm="giac")

[Out]

e^(-x^3/(x + 4) + 8*x^2*log(2)/(x + 4) + 16*x^2/(x + 4) - x*e^(2*e^(2*x))/(x + 4) + x/(x + 4))

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maple [A]  time = 0.50, size = 33, normalized size = 0.97

method	result	size
risch	\({\mathrm e}^{\frac {x \left (8 x \ln \relax (2)-x^{2}-{\mathrm e}^{2 \,{\mathrm e}^{2 x}}+16 x +1\right )}{4+x}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^2-16*x)*exp(x)^2-4)*exp(exp(x)^2)^2+2*(4*x^2+32*x)*ln(2)-2*x^3+4*x^2+128*x+4)*exp((-x*exp(exp(x)^2
)^2+8*x^2*ln(2)-x^3+16*x^2+x)/(4+x))/(x^2+8*x+16),x,method=_RETURNVERBOSE)

[Out]

exp(x*(8*x*ln(2)-x^2-exp(2*exp(2*x))+16*x+1)/(4+x))

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maxima [A]  time = 0.66, size = 57, normalized size = 1.68 \begin {gather*} \frac {1}{4294967296} \, e^{\left (-x^{2} + 8 \, x \log \relax (2) + 20 \, x + \frac {4 \, e^{\left (2 \, e^{\left (2 \, x\right )}\right )}}{x + 4} + \frac {128 \, \log \relax (2)}{x + 4} + \frac {316}{x + 4} - e^{\left (2 \, e^{\left (2 \, x\right )}\right )} - 79\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^2-16*x)*exp(x)^2-4)*exp(exp(x)^2)^2+2*(4*x^2+32*x)*log(2)-2*x^3+4*x^2+128*x+4)*exp((-x*exp(e
xp(x)^2)^2+8*x^2*log(2)-x^3+16*x^2+x)/(4+x))/(x^2+8*x+16),x, algorithm="maxima")

[Out]

1/4294967296*e^(-x^2 + 8*x*log(2) + 20*x + 4*e^(2*e^(2*x))/(x + 4) + 128*log(2)/(x + 4) + 316/(x + 4) - e^(2*e
^(2*x)) - 79)

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mupad [B]  time = 4.71, size = 59, normalized size = 1.74 \begin {gather*} 2^{\frac {8\,x^2}{x+4}}\,{\mathrm {e}}^{\frac {x}{x+4}}\,{\mathrm {e}}^{-\frac {x\,{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}}}{x+4}}\,{\mathrm {e}}^{-\frac {x^3}{x+4}}\,{\mathrm {e}}^{\frac {16\,x^2}{x+4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x + 8*x^2*log(2) + 16*x^2 - x^3 - x*exp(2*exp(2*x)))/(x + 4))*(128*x + 2*log(2)*(32*x + 4*x^2) - exp
(2*exp(2*x))*(exp(2*x)*(16*x + 4*x^2) + 4) + 4*x^2 - 2*x^3 + 4))/(8*x + x^2 + 16),x)

[Out]

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

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sympy [A]  time = 0.77, size = 32, normalized size = 0.94 \begin {gather*} e^{\frac {- x^{3} + 8 x^{2} \log {\relax (2 )} + 16 x^{2} - x e^{2 e^{2 x}} + x}{x + 4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**2-16*x)*exp(x)**2-4)*exp(exp(x)**2)**2+2*(4*x**2+32*x)*ln(2)-2*x**3+4*x**2+128*x+4)*exp((-x
*exp(exp(x)**2)**2+8*x**2*ln(2)-x**3+16*x**2+x)/(4+x))/(x**2+8*x+16),x)

[Out]

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

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