\(\int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} (-6 e^{2 x} x-6 x^2+e^x (-6 x-6 x^2))-x \log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})+(-e^x x-x \log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})+(3 e^x+3 \log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})) \log (e^x+\log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}))) \log (-x+3 \log (e^x+\log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})))}{-e^x x-x \log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})+(3 e^x+3 \log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}})) \log (e^x+\log (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}))} \, dx\) [6704]

Optimal result

Integrand size = 311, antiderivative size = 30 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-2 e^x x-e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )+x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )-\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx \\ & = \int \left (\frac {6 e^{\left (e^x+x\right )^2} \left (1+e^x\right ) x \left (e^x+x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}+\frac {-2 e^x x+x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )+e^x x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )+x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 e^x \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right ) \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}\right ) \, dx \\ & = 6 \int \frac {e^{\left (e^x+x\right )^2} \left (1+e^x\right ) x \left (e^x+x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+\int \frac {-2 e^x x+x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )+e^x x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )+x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 e^x \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right ) \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx \\ & = 6 \int \left (\frac {e^{x+\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}+\frac {e^{\left (e^x+x\right )^2} x \left (1+x-\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}-\frac {e^{\left (e^x+x\right )^2} x \left (x-\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (-1+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}\right ) \, dx+\int \frac {e^x \left (-2 x+\left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )\right )+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right ) \left (x+\left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx \\ & = 6 \int \frac {e^{x+\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x \left (1+x-\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x \left (x-\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (-1+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+\int \left (\frac {3 x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}+\frac {-2 x+x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}\right ) \, dx \\ & = 3 \int \frac {x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+6 \int \left (\frac {e^{\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}+\frac {e^{\left (e^x+x\right )^2} x^2}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}-\frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}\right ) \, dx-6 \int \left (-\frac {e^{\left (e^x+x\right )^2} x^2}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}+\frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}+\frac {e^{\left (e^x+x\right )^2} x^2 \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}-\frac {e^{\left (e^x+x\right )^2} x \log ^2\left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}\right ) \, dx+6 \int \frac {e^{x+\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+\int \frac {-2 x+x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx \\ & = 3 \int \frac {x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{x+\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x^2}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x^2}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x^2 \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x \log ^2\left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+\int \left (-\frac {2 x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )}+\log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )\right ) \, dx \\ & = -\left (2 \int \frac {x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx\right )+3 \int \frac {x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{x+\left (e^x+x\right )^2} x}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x^2}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x^2}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx-6 \int \frac {e^{\left (e^x+x\right )^2} x^2 \log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+6 \int \frac {e^{\left (e^x+x\right )^2} x \log ^2\left (5 e^{-e^{\left (e^x+x\right )^2}}\right )}{\left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right )} \, dx+\int \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \]

[In]

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

[Out]

x*Log[-x + 3*Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]]]

Maple [A] (verified)

Time = 218.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13

method	result	size
risch	\(\ln \left (3 \ln \left (\ln \left (5\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}}\right )+{\mathrm e}^{x}\right )-x \right ) x\)	\(34\)
parallelrisch	\(x \ln \left (3 \ln \left (\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}}\right )+{\mathrm e}^{x}\right )-x \right )\)	\(34\)

[In]

int((((3*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*ln(ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-
x*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)*x)*ln(3*ln(ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x)
-x*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+(-6*x*exp(x)^2+(-6*x^2-6*x)*exp(x)-6*x^2)*exp(exp(x)^2+2*exp(x)*x+x
^2)+2*exp(x)*x)/((3*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*ln(ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)
))+exp(x))-x*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)*x),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x + 3 \, \log \left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} + e^{x} + \log \left (5\right )\right )\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((((3*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+3*exp(x))*ln(ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2))
)+exp(x))-x*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))-exp(x)*x)*ln(3*ln(ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2
)))+exp(x))-x)-x*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+(-6*x*exp(x)**2+(-6*x**2-6*x)*exp(x)-6*x**2)*exp(ex
p(x)**2+2*exp(x)*x+x**2)+2*exp(x)*x)/((3*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+3*exp(x))*ln(ln(5/exp(exp(e
xp(x)**2+2*exp(x)*x+x**2)))+exp(x))-x*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))-exp(x)*x),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x + 3 \, \log \left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} + e^{x} + \log \left (5\right )\right )\right ) \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((6*(x^2 + x*e^(2*x) + (x^2 + x)*e^x)*e^(x^2 + 2*x*e^x + e^(2*x)) - 2*x*e^x + (x*e^x + x*log(5*e^(-e^
(x^2 + 2*x*e^x + e^(2*x)))) - 3*(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))*log(e^x + log(5*e^(-e^(x^2 + 2
*x*e^x + e^(2*x))))))*log(-x + 3*log(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))) + x*log(5*e^(-e^(x^2 + 2*
x*e^x + e^(2*x)))))/(x*e^x + x*log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))) - 3*(e^x + log(5*e^(-e^(x^2 + 2*x*e^x +
e^(2*x)))))*log(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))), x)

Mupad [B] (verification not implemented)

Time = 11.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.30 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=\frac {x\,\ln \left (5\right )\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}}+\frac {x\,{\mathrm {e}}^x\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}}-\frac {x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}} \]

[In]

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

[Out]

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