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

Optimal. Leaf size=15 \[ e^{e^{\frac {x}{-4-x}} x} \]

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Rubi [F]  time = 1.69, 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 (e^{\frac {-x+(4+x) \log (x)}{4+x}}+\frac {-x+(4+x) \log (x)}{4+x}\right ) \left (16+4 x+x^2\right )}{16 x+8 x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][E^(x/E^(x/(4 + x)) - x/(4 + x)), x] + 16*Defer[Int][E^(x/E^(x/(4 + x)) - x/(4 + x))/(4 + x)^2, x] -
4*Defer[Int][E^(x/E^(x/(4 + x)) - x/(4 + x))/(4 + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (e^{\frac {-x+(4+x) \log (x)}{4+x}}+\frac {-x+(4+x) \log (x)}{4+x}\right ) \left (16+4 x+x^2\right )}{x \left (16+8 x+x^2\right )} \, dx\\ &=\int \frac {\exp \left (e^{\frac {-x+(4+x) \log (x)}{4+x}}+\frac {-x+(4+x) \log (x)}{4+x}\right ) \left (16+4 x+x^2\right )}{x (4+x)^2} \, dx\\ &=\int \frac {e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}} \left (16+4 x+x^2\right )}{(4+x)^2} \, dx\\ &=\int \left (e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}}+\frac {16 e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}}}{(4+x)^2}-\frac {4 e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}}}{4+x}\right ) \, dx\\ &=-\left (4 \int \frac {e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}}}{4+x} \, dx\right )+16 \int \frac {e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}}}{(4+x)^2} \, dx+\int e^{e^{-\frac {x}{4+x}} x-\frac {x}{4+x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 15, normalized size = 1.00 \begin {gather*} e^{e^{-1+\frac {4}{4+x}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(E^(-1 + 4/(4 + x))*x)

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fricas [B]  time = 0.82, size = 56, normalized size = 3.73 \begin {gather*} e^{\left (\frac {{\left (x + 4\right )} e^{\left (\frac {{\left (x + 4\right )} \log \relax (x) - x}{x + 4}\right )} + {\left (x + 4\right )} \log \relax (x) - x}{x + 4} - \frac {{\left (x + 4\right )} \log \relax (x) - x}{x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+16)*exp(((4+x)*log(x)-x)/(4+x))*exp(exp(((4+x)*log(x)-x)/(4+x)))/(x^3+8*x^2+16*x),x, algori
thm="fricas")

[Out]

e^(((x + 4)*e^(((x + 4)*log(x) - x)/(x + 4)) + (x + 4)*log(x) - x)/(x + 4) - ((x + 4)*log(x) - x)/(x + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+16)*exp(((4+x)*log(x)-x)/(4+x))*exp(exp(((4+x)*log(x)-x)/(4+x)))/(x^3+8*x^2+16*x),x, algori
thm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 21, normalized size = 1.40

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {x \ln \relax (x )+4 \ln \relax (x )-x}{4+x}}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+4*x+16)*exp(((4+x)*ln(x)-x)/(4+x))*exp(exp(((4+x)*ln(x)-x)/(4+x)))/(x^3+8*x^2+16*x),x,method=_RETURNV
ERBOSE)

[Out]

exp(exp((x*ln(x)+4*ln(x)-x)/(4+x)))

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maxima [A]  time = 0.59, size = 13, normalized size = 0.87 \begin {gather*} e^{\left (x e^{\left (\frac {4}{x + 4} - 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+16)*exp(((4+x)*log(x)-x)/(4+x))*exp(exp(((4+x)*log(x)-x)/(4+x)))/(x^3+8*x^2+16*x),x, algori
thm="maxima")

[Out]

e^(x*e^(4/(x + 4) - 1))

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mupad [B]  time = 4.63, size = 12, normalized size = 0.80 \begin {gather*} {\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {x}{x+4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x*exp(-x/(x + 4)))

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

[Out]

Timed out

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