\(\int \frac {e^{e^x x} (e^5+x^2+e^x (x^3+x^4)+e^{2 x} (-e^5 x-2 x^3+e^x (-2 x^4-2 x^5))+e^{4 x} (x^4+e^x (x^5+x^6))+(-e^5+e^{5+x} (x+x^2)+e^{2 x} (e^5 (2 x+2 x^2)+e^{5+x} (-x^2-x^3))) \log (x))}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx\) [9727]

Optimal result

Integrand size = 162, antiderivative size = 32 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=e^{e^x x} \left (x-\frac {e^5 \log (x)}{-x+e^{2 x} x^2}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx \]

[In]

Int[(E^(E^x*x)*(E^5 + x^2 + E^x*(x^3 + x^4) + E^(2*x)*(-(E^5*x) - 2*x^3 + E^x*(-2*x^4 - 2*x^5)) + E^(4*x)*(x^4
+ E^x*(x^5 + x^6)) + (-E^5 + E^(5 + x)*(x + x^2) + E^(2*x)*(E^5*(2*x + 2*x^2) + E^(5 + x)*(-x^2 - x^3)))*Log[
x]))/(x^2 - 2*E^(2*x)*x^3 + E^(4*x)*x^4),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{e^x x} \left (-\left (\left (-1+e^{2 x} x\right ) \left (e^5+x^2-e^{2 x} x^3+e^x x^3 (1+x)-e^{3 x} x^4 (1+x)\right )\right )-e^5 \left (1-e^x x (1+x)-2 e^{2 x} x (1+x)+e^{3 x} x^2 (1+x)\right ) \log (x)\right )}{x^2 \left (1-e^{2 x} x\right )^2} \, dx \\ & = \int \left (e^{e^x x}+e^{x+e^x x} x (1+x)+\frac {e^{5+e^x x} (1+2 x) \log (x)}{x^2 \left (-1+e^{2 x} x\right )^2}-\frac {e^{5+e^x x} \left (1-2 \log (x)-2 x \log (x)+e^x x \log (x)+e^x x^2 \log (x)\right )}{x^2 \left (-1+e^{2 x} x\right )}\right ) \, dx \\ & = \int e^{e^x x} \, dx+\int e^{x+e^x x} x (1+x) \, dx+\int \frac {e^{5+e^x x} (1+2 x) \log (x)}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx-\int \frac {e^{5+e^x x} \left (1-2 \log (x)-2 x \log (x)+e^x x \log (x)+e^x x^2 \log (x)\right )}{x^2 \left (-1+e^{2 x} x\right )} \, dx \\ & = \log (x) \int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx+\int e^{e^x x} \, dx+\int \left (e^{x+e^x x} x+e^{x+e^x x} x^2\right ) \, dx-\int \frac {e^{5+e^x x} \left (-1-(1+x) \left (-2+e^x x\right ) \log (x)\right )}{x^2 \left (1-e^{2 x} x\right )} \, dx-\int \frac {\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx+2 \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx}{x} \, dx \\ & = \log (x) \int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx+\int e^{e^x x} \, dx+\int e^{x+e^x x} x \, dx+\int e^{x+e^x x} x^2 \, dx-\int \left (\frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )}+\frac {e^{5+x+e^x x} \log (x)}{-1+e^{2 x} x}-\frac {2 e^{5+e^x x} \log (x)}{x^2 \left (-1+e^{2 x} x\right )}-\frac {2 e^{5+e^x x} \log (x)}{x \left (-1+e^{2 x} x\right )}+\frac {e^{5+x+e^x x} \log (x)}{x \left (-1+e^{2 x} x\right )}\right ) \, dx-\int \left (\frac {\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx}{x}+\frac {2 \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx}{x}\right ) \, dx \\ & = 2 \int \frac {e^{5+e^x x} \log (x)}{x^2 \left (-1+e^{2 x} x\right )} \, dx+2 \int \frac {e^{5+e^x x} \log (x)}{x \left (-1+e^{2 x} x\right )} \, dx-2 \int \frac {\int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx}{x} \, dx+\log (x) \int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx+\int e^{e^x x} \, dx+\int e^{x+e^x x} x \, dx+\int e^{x+e^x x} x^2 \, dx-\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )} \, dx-\int \frac {e^{5+x+e^x x} \log (x)}{-1+e^{2 x} x} \, dx-\int \frac {e^{5+x+e^x x} \log (x)}{x \left (-1+e^{2 x} x\right )} \, dx-\int \frac {\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx}{x} \, dx \\ & = -\left (2 \int \frac {\int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx}{x} \, dx\right )-2 \int \frac {\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )} \, dx}{x} \, dx-2 \int \frac {\int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )} \, dx}{x} \, dx+\log (x) \int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx-\log (x) \int \frac {e^{5+x+e^x x}}{-1+e^{2 x} x} \, dx-\log (x) \int \frac {e^{5+x+e^x x}}{x \left (-1+e^{2 x} x\right )} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )^2} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )} \, dx+(2 \log (x)) \int \frac {e^{5+e^x x}}{x \left (-1+e^{2 x} x\right )} \, dx+\int e^{e^x x} \, dx+\int e^{x+e^x x} x \, dx+\int e^{x+e^x x} x^2 \, dx-\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )} \, dx-\int \frac {\int \frac {e^{5+e^x x}}{x^2 \left (-1+e^{2 x} x\right )^2} \, dx}{x} \, dx+\int \frac {\int \frac {e^{5+x+e^x x}}{-1+e^{2 x} x} \, dx}{x} \, dx+\int \frac {\int \frac {e^{5+x+e^x x}}{x \left (-1+e^{2 x} x\right )} \, dx}{x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=e^{e^x x} \left (x-\frac {e^5 \log (x)}{x \left (-1+e^{2 x} x\right )}\right ) \]

[In]

Integrate[(E^(E^x*x)*(E^5 + x^2 + E^x*(x^3 + x^4) + E^(2*x)*(-(E^5*x) - 2*x^3 + E^x*(-2*x^4 - 2*x^5)) + E^(4*x
)*(x^4 + E^x*(x^5 + x^6)) + (-E^5 + E^(5 + x)*(x + x^2) + E^(2*x)*(E^5*(2*x + 2*x^2) + E^(5 + x)*(-x^2 - x^3))
)*Log[x]))/(x^2 - 2*E^(2*x)*x^3 + E^(4*x)*x^4),x]

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22

\[-\frac {\left (-{\mathrm e}^{2 x} x^{3}+{\mathrm e}^{5} \ln \left (x \right )+x^{2}\right ) {\mathrm e}^{{\mathrm e}^{x} x}}{x \left (x \,{\mathrm e}^{2 x}-1\right )}\]

[In]

int(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x)*exp(5)*exp(x)-exp(5))*ln(x)+((x^6+x^5)*ex
p(x)+x^4)*exp(2*x)^2+((-2*x^5-2*x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(exp(x)*x
)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x)

[Out]

-(-exp(x)^2*x^3+exp(5)*ln(x)+x^2)/x/(x*exp(x)^2-1)*exp(exp(x)*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {{\left (x^{3} e^{\left (2 \, x + 10\right )} - x^{2} e^{10} - e^{15} \log \left (x\right )\right )} e^{\left (x e^{x}\right )}}{x^{2} e^{\left (2 \, x + 10\right )} - x e^{10}} \]

[In]

integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x)*exp(5)*exp(x)-exp(5))*log(x)+((x^6+
x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2*x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(e
xp(x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm="fricas")

[Out]

(x^3*e^(2*x + 10) - x^2*e^10 - e^15*log(x))*e^(x*e^x)/(x^2*e^(2*x + 10) - x*e^10)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {\left (x^{3} e^{2 x} - x^{2} - e^{5} \log {\left (x \right )}\right ) e^{x e^{x}}}{x^{2} e^{2 x} - x} \]

[In]

integrate(((((-x**3-x**2)*exp(5)*exp(x)+(2*x**2+2*x)*exp(5))*exp(2*x)+(x**2+x)*exp(5)*exp(x)-exp(5))*ln(x)+((x
**6+x**5)*exp(x)+x**4)*exp(2*x)**2+((-2*x**5-2*x**4)*exp(x)-x*exp(5)-2*x**3)*exp(2*x)+(x**4+x**3)*exp(x)+x**2+
exp(5))*exp(exp(x)*x)/(x**4*exp(2*x)**2-2*exp(2*x)*x**3+x**2),x)

[Out]

(x**3*exp(2*x) - x**2 - exp(5)*log(x))*exp(x*exp(x))/(x**2*exp(2*x) - x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.25 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {{\left (x^{3} e^{\left (2 \, x\right )} - x^{2} - e^{5} \log \left (x\right )\right )} e^{\left (x e^{x}\right )}}{x^{2} e^{\left (2 \, x\right )} - x} \]

[In]

integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x)*exp(5)*exp(x)-exp(5))*log(x)+((x^6+
x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2*x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(e
xp(x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm="maxima")

[Out]

(x^3*e^(2*x) - x^2 - e^5*log(x))*e^(x*e^x)/(x^2*e^(2*x) - x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\frac {x^{3} e^{\left (x e^{x} + 2 \, x\right )} - x^{2} e^{\left (x e^{x}\right )} - e^{\left (x e^{x} + 5\right )} \log \left (x\right )}{x^{2} e^{\left (2 \, x\right )} - x} \]

[In]

integrate(((((-x^3-x^2)*exp(5)*exp(x)+(2*x^2+2*x)*exp(5))*exp(2*x)+(x^2+x)*exp(5)*exp(x)-exp(5))*log(x)+((x^6+
x^5)*exp(x)+x^4)*exp(2*x)^2+((-2*x^5-2*x^4)*exp(x)-x*exp(5)-2*x^3)*exp(2*x)+(x^4+x^3)*exp(x)+x^2+exp(5))*exp(e
xp(x)*x)/(x^4*exp(2*x)^2-2*exp(2*x)*x^3+x^2),x, algorithm="giac")

[Out]

(x^3*e^(x*e^x + 2*x) - x^2*e^(x*e^x) - e^(x*e^x + 5)*log(x))/(x^2*e^(2*x) - x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{e^x x} \left (e^5+x^2+e^x \left (x^3+x^4\right )+e^{2 x} \left (-e^5 x-2 x^3+e^x \left (-2 x^4-2 x^5\right )\right )+e^{4 x} \left (x^4+e^x \left (x^5+x^6\right )\right )+\left (-e^5+e^{5+x} \left (x+x^2\right )+e^{2 x} \left (e^5 \left (2 x+2 x^2\right )+e^{5+x} \left (-x^2-x^3\right )\right )\right ) \log (x)\right )}{x^2-2 e^{2 x} x^3+e^{4 x} x^4} \, dx=\int \frac {{\mathrm {e}}^{x\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^5+{\mathrm {e}}^{4\,x}\,\left ({\mathrm {e}}^x\,\left (x^6+x^5\right )+x^4\right )+{\mathrm {e}}^x\,\left (x^4+x^3\right )-{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^x\,\left (2\,x^5+2\,x^4\right )+x\,{\mathrm {e}}^5+2\,x^3\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^{x+5}\,\left (x^2+x\right )-{\mathrm {e}}^5+{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^5\,\left (2\,x^2+2\,x\right )-{\mathrm {e}}^{x+5}\,\left (x^3+x^2\right )\right )\right )+x^2\right )}{x^4\,{\mathrm {e}}^{4\,x}-2\,x^3\,{\mathrm {e}}^{2\,x}+x^2} \,d x \]

[In]

int((exp(x*exp(x))*(exp(5) + exp(4*x)*(exp(x)*(x^5 + x^6) + x^4) + exp(x)*(x^3 + x^4) - exp(2*x)*(exp(x)*(2*x^
4 + 2*x^5) + x*exp(5) + 2*x^3) + log(x)*(exp(2*x)*(exp(5)*(2*x + 2*x^2) - exp(5)*exp(x)*(x^2 + x^3)) - exp(5)
+ exp(5)*exp(x)*(x + x^2)) + x^2))/(x^4*exp(4*x) - 2*x^3*exp(2*x) + x^2),x)

[Out]

int((exp(x*exp(x))*(exp(5) + exp(4*x)*(exp(x)*(x^5 + x^6) + x^4) + exp(x)*(x^3 + x^4) - exp(2*x)*(exp(x)*(2*x^
4 + 2*x^5) + x*exp(5) + 2*x^3) + log(x)*(exp(x + 5)*(x + x^2) - exp(5) + exp(2*x)*(exp(5)*(2*x + 2*x^2) - exp(
x + 5)*(x^2 + x^3))) + x^2))/(x^4*exp(4*x) - 2*x^3*exp(2*x) + x^2), x)