\(\int \frac {-5 x+5 e^x x+(-2+e^x (5-5 x)-3 x+5 x^2) \log (\frac {1}{5} (-2+5 e^x-5 x))+((2+3 x-5 x^2+e^x (-5+5 x)) \log (\frac {1}{5} (-2+5 e^x-5 x))+(2-5 e^x+5 x) \log (\frac {1}{5} (-2+5 e^x-5 x)) \log (x \log (\frac {1}{5} (-2+5 e^x-5 x)))) \log (-1+x-\log (x \log (\frac {1}{5} (-2+5 e^x-5 x))))}{((-2 x-3 x^2+5 x^3+e^x (5 x-5 x^2)) \log (\frac {1}{5} (-2+5 e^x-5 x))+(-2 x+5 e^x x-5 x^2) \log (\frac {1}{5} (-2+5 e^x-5 x)) \log (x \log (\frac {1}{5} (-2+5 e^x-5 x)))) \log (-1+x-\log (x \log (\frac {1}{5} (-2+5 e^x-5 x))))} \, dx\) [5405]

Optimal result

Integrand size = 272, antiderivative size = 26 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=-\log (x)+\log \left (\log \left (-1+x-\log \left (x \log \left (-\frac {2}{5}+e^x-x\right )\right )\right )\right ) \]

[Out]

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

Rubi [F]

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

[In]

Int[(-5*x + 5*E^x*x + (-2 + E^x*(5 - 5*x) - 3*x + 5*x^2)*Log[(-2 + 5*E^x - 5*x)/5] + ((2 + 3*x - 5*x^2 + E^x*(
-5 + 5*x))*Log[(-2 + 5*E^x - 5*x)/5] + (2 - 5*E^x + 5*x)*Log[(-2 + 5*E^x - 5*x)/5]*Log[x*Log[(-2 + 5*E^x - 5*x
)/5]])*Log[-1 + x - Log[x*Log[(-2 + 5*E^x - 5*x)/5]]])/(((-2*x - 3*x^2 + 5*x^3 + E^x*(5*x - 5*x^2))*Log[(-2 +
5*E^x - 5*x)/5] + (-2*x + 5*E^x*x - 5*x^2)*Log[(-2 + 5*E^x - 5*x)/5]*Log[x*Log[(-2 + 5*E^x - 5*x)/5]])*Log[-1
+ x - Log[x*Log[(-2 + 5*E^x - 5*x)/5]]]),x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=-\log (x)+\log \left (\log \left (-1+x-\log \left (x \log \left (-\frac {2}{5}+e^x-x\right )\right )\right )\right ) \]

[In]

Integrate[(-5*x + 5*E^x*x + (-2 + E^x*(5 - 5*x) - 3*x + 5*x^2)*Log[(-2 + 5*E^x - 5*x)/5] + ((2 + 3*x - 5*x^2 +
E^x*(-5 + 5*x))*Log[(-2 + 5*E^x - 5*x)/5] + (2 - 5*E^x + 5*x)*Log[(-2 + 5*E^x - 5*x)/5]*Log[x*Log[(-2 + 5*E^x
- 5*x)/5]])*Log[-1 + x - Log[x*Log[(-2 + 5*E^x - 5*x)/5]]])/(((-2*x - 3*x^2 + 5*x^3 + E^x*(5*x - 5*x^2))*Log[
(-2 + 5*E^x - 5*x)/5] + (-2*x + 5*E^x*x - 5*x^2)*Log[(-2 + 5*E^x - 5*x)/5]*Log[x*Log[(-2 + 5*E^x - 5*x)/5]])*L
og[-1 + x - Log[x*Log[(-2 + 5*E^x - 5*x)/5]]]),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x - \log \left (x \log \left (-x + e^{x} - \frac {2}{5}\right )\right ) - 1\right )\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate((((-5*exp(x)+5*x+2)*ln(exp(x)-x-2/5)*ln(x*ln(exp(x)-x-2/5))+((5*x-5)*exp(x)-5*x**2+3*x+2)*ln(exp(x)-
x-2/5))*ln(-ln(x*ln(exp(x)-x-2/5))+x-1)+((-5*x+5)*exp(x)+5*x**2-3*x-2)*ln(exp(x)-x-2/5)+5*exp(x)*x-5*x)/((5*ex
p(x)*x-5*x**2-2*x)*ln(exp(x)-x-2/5)*ln(x*ln(exp(x)-x-2/5))+((-5*x**2+5*x)*exp(x)+5*x**3-3*x**2-2*x)*ln(exp(x)-
x-2/5))/ln(-ln(x*ln(exp(x)-x-2/5))+x-1),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x - \log \left (x\right ) - \log \left (-\log \left (5\right ) + \log \left (-5 \, x + 5 \, e^{x} - 2\right )\right ) - 1\right )\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 2.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=-\log \left (x\right ) + \log \left (\log \left (x - \log \left (x\right ) - \log \left (-\log \left (5\right ) + \log \left (-5 \, x + 5 \, e^{x} - 2\right )\right ) - 1\right )\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-5 x+5 e^x x+\left (-2+e^x (5-5 x)-3 x+5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (\left (2+3 x-5 x^2+e^x (-5+5 x)\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (2-5 e^x+5 x\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )}{\left (\left (-2 x-3 x^2+5 x^3+e^x \left (5 x-5 x^2\right )\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )+\left (-2 x+5 e^x x-5 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right ) \log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right ) \log \left (-1+x-\log \left (x \log \left (\frac {1}{5} \left (-2+5 e^x-5 x\right )\right )\right )\right )} \, dx=\ln \left (\ln \left (x-\ln \left (x\,\ln \left ({\mathrm {e}}^x-x-\frac {2}{5}\right )\right )-1\right )\right )-\ln \left (x\right ) \]

[In]

int((5*x - log(x - log(x*log(exp(x) - x - 2/5)) - 1)*(log(exp(x) - x - 2/5)*(3*x + exp(x)*(5*x - 5) - 5*x^2 +
2) + log(exp(x) - x - 2/5)*log(x*log(exp(x) - x - 2/5))*(5*x - 5*exp(x) + 2)) - 5*x*exp(x) + log(exp(x) - x -
2/5)*(3*x + exp(x)*(5*x - 5) - 5*x^2 + 2))/(log(x - log(x*log(exp(x) - x - 2/5)) - 1)*(log(exp(x) - x - 2/5)*(
2*x - exp(x)*(5*x - 5*x^2) + 3*x^2 - 5*x^3) + log(exp(x) - x - 2/5)*log(x*log(exp(x) - x - 2/5))*(2*x - 5*x*ex
p(x) + 5*x^2))),x)

[Out]

log(log(x - log(x*log(exp(x) - x - 2/5)) - 1)) - log(x)