\(\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} (-10 e x-10 e^{2-x} x) \log (x)) \, dx\) [10328]

Optimal result

Integrand size = 112, antiderivative size = 29 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x^{1+\frac {e^{-2 e^{1+5 \left (-e^{1-x}+x\right )}}}{x^2}} \]

[Out]

exp(ln(x)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2)*x

Rubi [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx \]

[In]

Int[(x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))*(1 + E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2 - 2*Log[x] + E^
(-5*E^(1 - x) + 5*x)*(-10*E*x - 10*E^(2 - x)*x)*Log[x]))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)),x]

[Out]

Defer[Int][x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)), x] - 2*Log[x]*Def
er[Int][x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)), x] - 10*Log[x]*Defer
[Int][E^(2 - 5*E^(1 - x) - 2*E^(1 - 5*E^(1 - x) + 5*x) + 4*x)*x^(-1 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))
, x] - 10*Log[x]*Defer[Int][E^(1 - 5*E^(1 - x) - 2*E^(1 - 5*E^(1 - x) + 5*x) + 5*x)*x^(-1 + 1/(E^(2*E^(1 - 5*E
^(1 - x) + 5*x))*x^2)), x] + Defer[Int][x^(1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2)), x] + 2*Defer[Int][Defer[I
nt][x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)), x]/x, x] + 10*Defer[Int]
[Defer[Int][E^(2 - 5*E^(1 - x) - 2*E^(1 - 5*E^(1 - x) + 5*x) + 4*x)*x^(-1 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))
*x^2)), x]/x, x] + 10*Defer[Int][Defer[Int][E^(1 - 5*E^(1 - x) - 2*E^(1 - 5*E^(1 - x) + 5*x) + 5*x)*x^(-1 + 1/
(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2)), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}}+x^{\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}}-2 e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \log (x)-10 e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} \left (e+e^x\right ) x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \log (x)\right ) \, dx \\ & = -\left (2 \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \log (x) \, dx\right )-10 \int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} \left (e+e^x\right ) x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \log (x) \, dx+\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int x^{\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx \\ & = 2 \int \frac {\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx+10 \int \frac {\int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx-(2 \log (x)) \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int x^{\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx \\ & = 2 \int \frac {\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx+10 \int \left (\frac {\int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x}+\frac {\int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x}\right ) \, dx-(2 \log (x)) \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int x^{\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx \\ & = 2 \int \frac {\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx+10 \int \frac {\int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx+10 \int \frac {\int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx}{x} \, dx-(2 \log (x)) \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{2-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+4 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx-(10 \log (x)) \int e^{1-5 e^{1-x}-2 e^{1-5 e^{1-x}+5 x}+5 x} x^{-1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx+\int x^{\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x^{1+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \]

[In]

Integrate[(x^(-2 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))*(1 + E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2 - 2*Log[x
] + E^(-5*E^(1 - x) + 5*x)*(-10*E*x - 10*E^(2 - x)*x)*Log[x]))/E^(2*E^(1 - 5*E^(1 - x) + 5*x)),x]

[Out]

x^(1 + 1/(E^(2*E^(1 - 5*E^(1 - x) + 5*x))*x^2))

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90

\[x^{\frac {{\mathrm e}^{-2 \,{\mathrm e}^{1-5 \,{\mathrm e}^{1-x}+5 x}}}{x^{2}}} x\]

[In]

int((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*x*exp(1))*ln(x)*exp(-5*exp(1-x)+5*x)-2*l
n(x)+1)*exp(ln(x)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x)

[Out]

x^(1/x^2*exp(-2*exp(1-5*exp(1-x)+5*x)))*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=x x^{\frac {e^{\left (-2 \, e^{\left ({\left ({\left (5 \, x + 1\right )} e - 5 \, e^{\left (-x + 2\right )}\right )} e^{\left (-1\right )}\right )}\right )}}{x^{2}}} \]

[In]

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*x*exp(1))*log(x)*exp(-5*exp(1-x)+5
*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, a
lgorithm="fricas")

[Out]

x*x^(e^(-2*e^(((5*x + 1)*e - 5*e^(-x + 2))*e^(-1)))/x^2)

Sympy [F(-1)]

Timed out. \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\text {Timed out} \]

[In]

integrate((x**2*exp(exp(1)*exp(-5*exp(1-x)+5*x))**2+(-10*x*exp(1)*exp(1-x)-10*x*exp(1))*ln(x)*exp(-5*exp(1-x)+
5*x)-2*ln(x)+1)*exp(ln(x)/x**2/exp(exp(1)*exp(-5*exp(1-x)+5*x))**2)/x**2/exp(exp(1)*exp(-5*exp(1-x)+5*x))**2,x
)

[Out]

Timed out

Maxima [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int { \frac {{\left (x^{2} e^{\left (2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )} - 10 \, {\left (x e + x e^{\left (-x + 2\right )}\right )} e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )}\right )} \log \left (x\right ) - 2 \, \log \left (x\right ) + 1\right )} x^{\frac {e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}}} e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}} \,d x } \]

[In]

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*x*exp(1))*log(x)*exp(-5*exp(1-x)+5
*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, a
lgorithm="maxima")

[Out]

integrate((x^2*e^(2*e^(5*x - 5*e^(-x + 1) + 1)) - 10*(x*e + x*e^(-x + 2))*e^(5*x - 5*e^(-x + 1))*log(x) - 2*lo
g(x) + 1)*x^(e^(-2*e^(5*x - 5*e^(-x + 1) + 1))/x^2 - 2)*e^(-2*e^(5*x - 5*e^(-x + 1) + 1)), x)

Giac [F]

\[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int { \frac {{\left (x^{2} e^{\left (2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )} - 10 \, {\left (x e + x e^{\left (-x + 2\right )}\right )} e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )}\right )} \log \left (x\right ) - 2 \, \log \left (x\right ) + 1\right )} x^{\frac {e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}}} e^{\left (-2 \, e^{\left (5 \, x - 5 \, e^{\left (-x + 1\right )} + 1\right )}\right )}}{x^{2}} \,d x } \]

[In]

integrate((x^2*exp(exp(1)*exp(-5*exp(1-x)+5*x))^2+(-10*x*exp(1)*exp(1-x)-10*x*exp(1))*log(x)*exp(-5*exp(1-x)+5
*x)-2*log(x)+1)*exp(log(x)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2)/x^2/exp(exp(1)*exp(-5*exp(1-x)+5*x))^2,x, a
lgorithm="giac")

[Out]

integrate((x^2*e^(2*e^(5*x - 5*e^(-x + 1) + 1)) - 10*(x*e + x*e^(-x + 2))*e^(5*x - 5*e^(-x + 1))*log(x) - 2*lo
g(x) + 1)*x^(e^(-2*e^(5*x - 5*e^(-x + 1) + 1))/x^2)*e^(-2*e^(5*x - 5*e^(-x + 1) + 1))/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int e^{-2 e^{1-5 e^{1-x}+5 x}} x^{-2+\frac {e^{-2 e^{1-5 e^{1-x}+5 x}}}{x^2}} \left (1+e^{2 e^{1-5 e^{1-x}+5 x}} x^2-2 \log (x)+e^{-5 e^{1-x}+5 x} \left (-10 e x-10 e^{2-x} x\right ) \log (x)\right ) \, dx=\int -\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{-2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}\,\ln \left (x\right )}{x^2}}\,{\mathrm {e}}^{-2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}\,\left (2\,\ln \left (x\right )-x^2\,{\mathrm {e}}^{2\,\mathrm {e}\,{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}}+{\mathrm {e}}^{5\,x-5\,{\mathrm {e}}^{1-x}}\,\ln \left (x\right )\,\left (10\,x\,\mathrm {e}+10\,x\,\mathrm {e}\,{\mathrm {e}}^{1-x}\right )-1\right )}{x^2} \,d x \]

[In]

int(-(exp((exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*log(x))/x^2)*exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*(2*log(x
) - x^2*exp(2*exp(1)*exp(5*x - 5*exp(1 - x))) + exp(5*x - 5*exp(1 - x))*log(x)*(10*x*exp(1) + 10*x*exp(1)*exp(
1 - x)) - 1))/x^2,x)

[Out]

int(-(exp((exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*log(x))/x^2)*exp(-2*exp(1)*exp(5*x - 5*exp(1 - x)))*(2*log(x
) - x^2*exp(2*exp(1)*exp(5*x - 5*exp(1 - x))) + exp(5*x - 5*exp(1 - x))*log(x)*(10*x*exp(1) + 10*x*exp(1)*exp(
1 - x)) - 1))/x^2, x)