\(\int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} (5 e^x+e^{e^{e^{16 e^{-x}}}} (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x))) \, dx\) [9020]

Optimal result

Integrand size = 71, antiderivative size = 17 \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=x^{5+e^{e^{e^{16 e^{-x}}}}} \]

[Out]

exp((5+exp(exp(exp(4/exp(x))^4)))*ln(x))

Rubi [F]

\[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=\int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx \]

[In]

Int[(x^(4 + E^E^E^(16/E^x))*(5*E^x + E^E^E^(16/E^x)*(E^x - 16*E^(E^(16/E^x) + 16/E^x)*x*Log[x])))/E^x,x]

[Out]

5*Defer[Int][x^(4 + E^E^E^(16/E^x)), x] + Defer[Int][E^E^E^(16/E^x)*x^(4 + E^E^E^(16/E^x)), x] - 16*Log[x]*Def
er[Int][E^(E^E^(16/E^x) + (16 + E^(16/E^x + x))/E^x - x)*x^(5 + E^E^E^(16/E^x)), x] + 16*Defer[Int][Defer[Int]
[E^(E^E^(16/E^x) + E^(16/E^x) + 16/E^x - x)*x^(5 + E^E^E^(16/E^x)), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (5 x^{4+e^{e^{e^{16 e^{-x}}}}}+e^{e^{e^{16 e^{-x}}}-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx \\ & = 5 \int x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx+\int e^{e^{e^{16 e^{-x}}}-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right ) \, dx \\ & = 5 \int x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx+\int \left (e^{e^{e^{16 e^{-x}}}} x^{4+e^{e^{e^{16 e^{-x}}}}}-16 \exp \left (e^{e^{16 e^{-x}}}+e^{-x} \left (16+e^{16 e^{-x}+x}\right )-x\right ) x^{5+e^{e^{e^{16 e^{-x}}}}} \log (x)\right ) \, dx \\ & = 5 \int x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx-16 \int \exp \left (e^{e^{16 e^{-x}}}+e^{-x} \left (16+e^{16 e^{-x}+x}\right )-x\right ) x^{5+e^{e^{e^{16 e^{-x}}}}} \log (x) \, dx+\int e^{e^{e^{16 e^{-x}}}} x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx \\ & = 5 \int x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx+16 \int \frac {\int e^{e^{e^{16 e^{-x}}}+e^{16 e^{-x}}+16 e^{-x}-x} x^{5+e^{e^{e^{16 e^{-x}}}}} \, dx}{x} \, dx-(16 \log (x)) \int \exp \left (e^{e^{16 e^{-x}}}+e^{-x} \left (16+e^{16 e^{-x}+x}\right )-x\right ) x^{5+e^{e^{e^{16 e^{-x}}}}} \, dx+\int e^{e^{e^{16 e^{-x}}}} x^{4+e^{e^{e^{16 e^{-x}}}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=x^{5+e^{e^{e^{16 e^{-x}}}}} \]

[In]

Integrate[(x^(4 + E^E^E^(16/E^x))*(5*E^x + E^E^E^(16/E^x)*(E^x - 16*E^(E^(16/E^x) + 16/E^x)*x*Log[x])))/E^x,x]

[Out]

x^(5 + E^E^E^(16/E^x))

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[x^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{16 \,{\mathrm e}^{-x}}}}} x^{5}\]

[In]

int(((-16*x*ln(x)*exp(4/exp(x))^4*exp(exp(4/exp(x))^4)+exp(x))*exp(exp(exp(4/exp(x))^4))+5*exp(x))*exp(ln(x)*e
xp(exp(exp(4/exp(x))^4))+5*ln(x))/exp(x)/x,x)

[Out]

x^exp(exp(exp(16*exp(-x))))*x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=e^{\left (e^{\left (e^{\left ({\left (e^{\left (x + 16 \, e^{\left (-x\right )}\right )} + 16\right )} e^{\left (-x\right )} - 16 \, e^{\left (-x\right )}\right )}\right )} \log \left (x\right ) + 5 \, \log \left (x\right )\right )} \]

[In]

integrate(((-16*x*log(x)*exp(4/exp(x))^4*exp(exp(4/exp(x))^4)+exp(x))*exp(exp(exp(4/exp(x))^4))+5*exp(x))*exp(
log(x)*exp(exp(exp(4/exp(x))^4))+5*log(x))/exp(x)/x,x, algorithm="fricas")

[Out]

e^(e^(e^((e^(x + 16*e^(-x)) + 16)*e^(-x) - 16*e^(-x)))*log(x) + 5*log(x))

Sympy [F(-1)]

Timed out. \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=\text {Timed out} \]

[In]

integrate(((-16*x*ln(x)*exp(4/exp(x))**4*exp(exp(4/exp(x))**4)+exp(x))*exp(exp(exp(4/exp(x))**4))+5*exp(x))*ex
p(ln(x)*exp(exp(exp(4/exp(x))**4))+5*ln(x))/exp(x)/x,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=x^{5} x^{e^{\left (e^{\left (e^{\left (16 \, e^{\left (-x\right )}\right )}\right )}\right )}} \]

[In]

integrate(((-16*x*log(x)*exp(4/exp(x))^4*exp(exp(4/exp(x))^4)+exp(x))*exp(exp(exp(4/exp(x))^4))+5*exp(x))*exp(
log(x)*exp(exp(exp(4/exp(x))^4))+5*log(x))/exp(x)/x,x, algorithm="maxima")

[Out]

x^5*x^e^(e^(e^(16*e^(-x))))

Giac [F]

\[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=\int { -\frac {{\left ({\left (16 \, x e^{\left (16 \, e^{\left (-x\right )} + e^{\left (16 \, e^{\left (-x\right )}\right )}\right )} \log \left (x\right ) - e^{x}\right )} e^{\left (e^{\left (e^{\left (16 \, e^{\left (-x\right )}\right )}\right )}\right )} - 5 \, e^{x}\right )} e^{\left (e^{\left (e^{\left (e^{\left (16 \, e^{\left (-x\right )}\right )}\right )}\right )} \log \left (x\right ) - x + 5 \, \log \left (x\right )\right )}}{x} \,d x } \]

[In]

integrate(((-16*x*log(x)*exp(4/exp(x))^4*exp(exp(4/exp(x))^4)+exp(x))*exp(exp(exp(4/exp(x))^4))+5*exp(x))*exp(
log(x)*exp(exp(exp(4/exp(x))^4))+5*log(x))/exp(x)/x,x, algorithm="giac")

[Out]

integrate(-((16*x*e^(16*e^(-x) + e^(16*e^(-x)))*log(x) - e^x)*e^(e^(e^(16*e^(-x)))) - 5*e^x)*e^(e^(e^(e^(16*e^
(-x))))*log(x) - x + 5*log(x))/x, x)

Mupad [B] (verification not implemented)

Time = 13.59 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{-x} x^{4+e^{e^{e^{16 e^{-x}}}}} \left (5 e^x+e^{e^{e^{16 e^{-x}}}} \left (e^x-16 e^{e^{16 e^{-x}}+16 e^{-x}} x \log (x)\right )\right ) \, dx=x^{{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{16\,{\mathrm {e}}^{-x}}}}}\,x^5 \]

[In]

int((exp(5*log(x) + exp(exp(exp(16*exp(-x))))*log(x))*exp(-x)*(5*exp(x) + exp(exp(exp(16*exp(-x))))*(exp(x) -
16*x*exp(16*exp(-x))*exp(exp(16*exp(-x)))*log(x))))/x,x)

[Out]

x^exp(exp(exp(16*exp(-x))))*x^5