\(\int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} (e^x+(-2 x^2+x^3) \log (x))+(-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)) \log (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}) \log (\log (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}))}{(e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)) \log (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)})} \, dx\) [9505]

Optimal result

Integrand size = 212, antiderivative size = 33 \[ \int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} \left (e^x+\left (-2 x^2+x^3\right ) \log (x)\right )+\left (-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right ) \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right )}{\left (e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )} \, dx=-2-x \log \left (\log \left (\frac {1}{5} \left (x+\frac {e^{e^{-x} x^2}-x}{\log (x)}\right )\right )\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

-2*Defer[Int][x^2/(E^x*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] + Defer[Int][x^3/(E^x*Log[(E^(x^2/E^x
) - x + x*Log[x])/(5*Log[x])]), x] + Defer[Int][1/(Log[x]*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] -
2*Defer[Int][x^3/(E^x*(E^(x^2/E^x) - x + x*Log[x])*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] + Defer[I
nt][x^4/(E^x*(E^(x^2/E^x) - x + x*Log[x])*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] - Defer[Int][(x*Lo
g[x])/((E^(x^2/E^x) - x + x*Log[x])*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] + 2*Defer[Int][(x^3*Log[
x])/(E^x*(E^(x^2/E^x) - x + x*Log[x])*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] - Defer[Int][(x^4*Log[
x])/(E^x*(E^(x^2/E^x) - x + x*Log[x])*Log[(E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]), x] - Defer[Int][Log[Log[(
E^(x^2/E^x) - x + x*Log[x])/(5*Log[x])]], x]

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} \left (e^x+\left (-2 x^2+x^3\right ) \log (x)\right )+\left (-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right ) \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right )}{\left (e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )} \, dx=-x \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right ) \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 4.30

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

[In]

int(((-exp(x)*ln(x)*exp(x^2/exp(x))-x*exp(x)*ln(x)^2+x*exp(x)*ln(x))*ln(1/5*(exp(x^2/exp(x))+x*ln(x)-x)/ln(x))
*ln(ln(1/5*(exp(x^2/exp(x))+x*ln(x)-x)/ln(x)))+((x^3-2*x^2)*ln(x)+exp(x))*exp(x^2/exp(x))-x*exp(x)*ln(x)^2+x*e
xp(x)*ln(x)-exp(x)*x)/(exp(x)*ln(x)*exp(x^2/exp(x))+x*exp(x)*ln(x)^2-x*exp(x)*ln(x))/ln(1/5*(exp(x^2/exp(x))+x
*ln(x)-x)/ln(x)),x)

[Out]

-x*ln(-ln(5)-ln(ln(x))+ln((ln(x)-1)*x+exp(x^2*exp(-x)))-1/2*I*Pi*csgn(I/ln(x)*((ln(x)-1)*x+exp(x^2*exp(-x))))*
(-csgn(I/ln(x)*((ln(x)-1)*x+exp(x^2*exp(-x))))+csgn(I/ln(x)))*(-csgn(I/ln(x)*((ln(x)-1)*x+exp(x^2*exp(-x))))+c
sgn(I*((ln(x)-1)*x+exp(x^2*exp(-x))))))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} \left (e^x+\left (-2 x^2+x^3\right ) \log (x)\right )+\left (-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right ) \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right )}{\left (e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )} \, dx=-x \log \left (\log \left (\frac {{\left (x e^{x} \log \left (x\right ) - x e^{x} + e^{\left ({\left (x^{2} + x e^{x}\right )} e^{\left (-x\right )}\right )}\right )} e^{\left (-x\right )}}{5 \, \log \left (x\right )}\right )\right ) \]

[In]

integrate(((-exp(x)*log(x)*exp(x^2/exp(x))-x*exp(x)*log(x)^2+x*exp(x)*log(x))*log(1/5*(exp(x^2/exp(x))+x*log(x
)-x)/log(x))*log(log(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)))+((x^3-2*x^2)*log(x)+exp(x))*exp(x^2/exp(x))-x*e
xp(x)*log(x)^2+x*exp(x)*log(x)-exp(x)*x)/(exp(x)*log(x)*exp(x^2/exp(x))+x*exp(x)*log(x)^2-x*exp(x)*log(x))/log
(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(((-exp(x)*ln(x)*exp(x**2/exp(x))-x*exp(x)*ln(x)**2+x*exp(x)*ln(x))*ln(1/5*(exp(x**2/exp(x))+x*ln(x)-
x)/ln(x))*ln(ln(1/5*(exp(x**2/exp(x))+x*ln(x)-x)/ln(x)))+((x**3-2*x**2)*ln(x)+exp(x))*exp(x**2/exp(x))-x*exp(x
)*ln(x)**2+x*exp(x)*ln(x)-exp(x)*x)/(exp(x)*ln(x)*exp(x**2/exp(x))+x*exp(x)*ln(x)**2-x*exp(x)*ln(x))/ln(1/5*(e
xp(x**2/exp(x))+x*ln(x)-x)/ln(x)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} \left (e^x+\left (-2 x^2+x^3\right ) \log (x)\right )+\left (-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right ) \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right )}{\left (e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )} \, dx=-x \log \left (-\log \left (5\right ) + \log \left (x \log \left (x\right ) - x + e^{\left (x^{2} e^{\left (-x\right )}\right )}\right ) - \log \left (\log \left (x\right )\right )\right ) \]

[In]

integrate(((-exp(x)*log(x)*exp(x^2/exp(x))-x*exp(x)*log(x)^2+x*exp(x)*log(x))*log(1/5*(exp(x^2/exp(x))+x*log(x
)-x)/log(x))*log(log(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)))+((x^3-2*x^2)*log(x)+exp(x))*exp(x^2/exp(x))-x*e
xp(x)*log(x)^2+x*exp(x)*log(x)-exp(x)*x)/(exp(x)*log(x)*exp(x^2/exp(x))+x*exp(x)*log(x)^2-x*exp(x)*log(x))/log
(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

integrate(((-exp(x)*log(x)*exp(x^2/exp(x))-x*exp(x)*log(x)^2+x*exp(x)*log(x))*log(1/5*(exp(x^2/exp(x))+x*log(x
)-x)/log(x))*log(log(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)))+((x^3-2*x^2)*log(x)+exp(x))*exp(x^2/exp(x))-x*e
xp(x)*log(x)^2+x*exp(x)*log(x)-exp(x)*x)/(exp(x)*log(x)*exp(x^2/exp(x))+x*exp(x)*log(x)^2-x*exp(x)*log(x))/log
(1/5*(exp(x^2/exp(x))+x*log(x)-x)/log(x)),x, algorithm="giac")

[Out]

integrate(-(x*e^x*log(x)^2 - x*e^x*log(x) + (x*e^x*log(x)^2 - x*e^x*log(x) + e^(x^2*e^(-x) + x)*log(x))*log(1/
5*(x*log(x) - x + e^(x^2*e^(-x)))/log(x))*log(log(1/5*(x*log(x) - x + e^(x^2*e^(-x)))/log(x))) - ((x^3 - 2*x^2
)*log(x) + e^x)*e^(x^2*e^(-x)) + x*e^x)/((x*e^x*log(x)^2 - x*e^x*log(x) + e^(x^2*e^(-x) + x)*log(x))*log(1/5*(
x*log(x) - x + e^(x^2*e^(-x)))/log(x))), x)

Mupad [B] (verification not implemented)

Time = 15.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-e^x x+e^x x \log (x)-e^x x \log ^2(x)+e^{e^{-x} x^2} \left (e^x+\left (-2 x^2+x^3\right ) \log (x)\right )+\left (-e^{x+e^{-x} x^2} \log (x)+e^x x \log (x)-e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right ) \log \left (\log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )\right )}{\left (e^{x+e^{-x} x^2} \log (x)-e^x x \log (x)+e^x x \log ^2(x)\right ) \log \left (\frac {e^{e^{-x} x^2}-x+x \log (x)}{5 \log (x)}\right )} \, dx=-x\,\ln \left (\ln \left (\frac {{\mathrm {e}}^{x^2\,{\mathrm {e}}^{-x}}-x+x\,\ln \left (x\right )}{5\,\ln \left (x\right )}\right )\right ) \]

[In]

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

[Out]

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