\(\int \frac {4 e^5-5 x+(-e^5 x^2+x^3) \log (9 e^5 x^4-9 x^5)+(-e^5+x) \log (9 e^5 x^4-9 x^5) \log (\frac {1}{5} \log (9 e^5 x^4-9 x^5))}{(-4 x^3-4 x^4-x^5+e^5 (4 x^2+4 x^3+x^4)) \log (9 e^5 x^4-9 x^5)+(4 x^2+2 x^3+e^5 (-4 x-2 x^2)) \log (9 e^5 x^4-9 x^5) \log (\frac {1}{5} \log (9 e^5 x^4-9 x^5))+(e^5-x) \log (9 e^5 x^4-9 x^5) \log ^2(\frac {1}{5} \log (9 e^5 x^4-9 x^5))} \, dx\) [5765]

Optimal result

Integrand size = 239, antiderivative size = 30 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x (2+x)-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \]

[Out]

x/(x*(2+x)-ln(1/5*ln(9*x^4*(exp(5)-x))))

Rubi [F]

\[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx \]

[In]

Int[(4*E^5 - 5*x + (-(E^5*x^2) + x^3)*Log[9*E^5*x^4 - 9*x^5] + (-E^5 + x)*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5
*x^4 - 9*x^5]/5])/((-4*x^3 - 4*x^4 - x^5 + E^5*(4*x^2 + 4*x^3 + x^4))*Log[9*E^5*x^4 - 9*x^5] + (4*x^2 + 2*x^3
+ E^5*(-4*x - 2*x^2))*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5] + (E^5 - x)*Log[9*E^5*x^4 - 9*x^5]*
Log[Log[9*E^5*x^4 - 9*x^5]/5]^2),x]

[Out]

2*E^5*Defer[Int][(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^(-2), x] - 2*E^10*Defer[Int][(2*x + x^2 - Log[Log[9
*(E^5 - x)*x^4]/5])^(-2), x] - 2*E^5*(1 - E^5)*Defer[Int][(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^(-2), x] -
2*E^10*Defer[Int][1/((E^5 - x)*(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2), x] + 2*E^15*Defer[Int][1/((E^5 -
x)*(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2), x] + 2*E^10*(1 - E^5)*Defer[Int][1/((E^5 - x)*(2*x + x^2 - L
og[Log[9*(E^5 - x)*x^4]/5])^2), x] - 2*E^5*Defer[Int][x/(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2, x] - 2*(1
- E^5)*Defer[Int][x/(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2, x] - 2*Defer[Int][x^2/(2*x + x^2 - Log[Log[9
*(E^5 - x)*x^4]/5])^2, x] + 5*Defer[Int][1/(Log[9*(E^5 - x)*x^4]*(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2),
x] - E^5*Defer[Int][1/((E^5 - x)*Log[9*(E^5 - x)*x^4]*(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^2), x] + Defe
r[Int][(2*x + x^2 - Log[Log[9*(E^5 - x)*x^4]/5])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 e^5-5 x-\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (x^2+\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (x (2+x)-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx \\ & = \int \left (\frac {4 e^5-5 x-2 e^5 x \log \left (9 \left (e^5-x\right ) x^4\right )+2 \left (1-e^5\right ) x^2 \log \left (9 \left (e^5-x\right ) x^4\right )+2 x^3 \log \left (9 \left (e^5-x\right ) x^4\right )}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )}\right ) \, dx \\ & = \int \frac {4 e^5-5 x-2 e^5 x \log \left (9 \left (e^5-x\right ) x^4\right )+2 \left (1-e^5\right ) x^2 \log \left (9 \left (e^5-x\right ) x^4\right )+2 x^3 \log \left (9 \left (e^5-x\right ) x^4\right )}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx \\ & = \int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx+\int \frac {4 e^5-5 x-2 \left (e^5-x\right ) x (1+x) \log \left (9 \left (e^5-x\right ) x^4\right )}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (x (2+x)-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx \\ & = \int \left (-\frac {2 e^5 x}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {2 \left (1-e^5\right ) x^2}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {2 x^3}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {4 e^5}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}-\frac {5 x}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}\right ) \, dx+\int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx \\ & = 2 \int \frac {x^3}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-5 \int \frac {x}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 e^5\right ) \int \frac {x}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (4 e^5\right ) \int \frac {1}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (2 \left (1-e^5\right )\right ) \int \frac {x^2}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx \\ & = 2 \int \left (-\frac {e^{10}}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {e^{15}}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}-\frac {e^5 x}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}-\frac {x^2}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}\right ) \, dx-5 \int \left (-\frac {1}{\log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {e^5}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}\right ) \, dx-\left (2 e^5\right ) \int \left (-\frac {1}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {e^5}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}\right ) \, dx+\left (4 e^5\right ) \int \frac {1}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (2 \left (1-e^5\right )\right ) \int \left (-\frac {e^5}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}+\frac {e^{10}}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}-\frac {x}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2}\right ) \, dx+\int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx \\ & = -\left (2 \int \frac {x^2}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx\right )+5 \int \frac {1}{\log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (2 e^5\right ) \int \frac {1}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 e^5\right ) \int \frac {x}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (4 e^5\right ) \int \frac {1}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (5 e^5\right ) \int \frac {1}{\left (e^5-x\right ) \log \left (9 \left (e^5-x\right ) x^4\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 e^{10}\right ) \int \frac {1}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 e^{10}\right ) \int \frac {1}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (2 e^{15}\right ) \int \frac {1}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 \left (1-e^5\right )\right ) \int \frac {x}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx-\left (2 e^5 \left (1-e^5\right )\right ) \int \frac {1}{\left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\left (2 e^{10} \left (1-e^5\right )\right ) \int \frac {1}{\left (e^5-x\right ) \left (2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )\right )^2} \, dx+\int \frac {1}{2 x+x^2-\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=-\frac {x}{-2 x-x^2+\log \left (\frac {1}{5} \log \left (9 \left (e^5-x\right ) x^4\right )\right )} \]

[In]

Integrate[(4*E^5 - 5*x + (-(E^5*x^2) + x^3)*Log[9*E^5*x^4 - 9*x^5] + (-E^5 + x)*Log[9*E^5*x^4 - 9*x^5]*Log[Log
[9*E^5*x^4 - 9*x^5]/5])/((-4*x^3 - 4*x^4 - x^5 + E^5*(4*x^2 + 4*x^3 + x^4))*Log[9*E^5*x^4 - 9*x^5] + (4*x^2 +
2*x^3 + E^5*(-4*x - 2*x^2))*Log[9*E^5*x^4 - 9*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5] + (E^5 - x)*Log[9*E^5*x^4 - 9
*x^5]*Log[Log[9*E^5*x^4 - 9*x^5]/5]^2),x]

[Out]

-(x/(-2*x - x^2 + Log[Log[9*(E^5 - x)*x^4]/5]))

Maple [A] (verified)

Time = 4.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(\frac {x}{x^{2}+2 x -\ln \left (\frac {\ln \left (9 x^{4} \left ({\mathrm e}^{5}-x \right )\right )}{5}\right )}\)	\(29\)

[In]

int(((-exp(5)+x)*ln(9*x^4*exp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))+(-x^2*exp(5)+x^3)*ln(9*x^4*exp(5)-9*x^5
)+4*exp(5)-5*x)/((exp(5)-x)*ln(9*x^4*exp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)*exp(5)+2*x^3
+4*x^2)*ln(9*x^4*exp(5)-9*x^5)*ln(1/5*ln(9*x^4*exp(5)-9*x^5))+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*x^4-4*x^3)*ln(9*
x^4*exp(5)-9*x^5)),x,method=_RETURNVERBOSE)

[Out]

x/(x^2+2*x-ln(1/5*ln(9*x^4*(exp(5)-x))))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x - \log \left (\frac {1}{5} \, \log \left (-9 \, x^{5} + 9 \, x^{4} e^{5}\right )\right )} \]

[In]

integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*ex
p(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)
*exp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*
x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, algorithm="fricas")

[Out]

x/(x^2 + 2*x - log(1/5*log(-9*x^5 + 9*x^4*e^5)))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=- \frac {x}{- x^{2} - 2 x + \log {\left (\frac {\log {\left (- 9 x^{5} + 9 x^{4} e^{5} \right )}}{5} \right )}} \]

[In]

integrate(((-exp(5)+x)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp(5)-9*x**5))+(-x**2*exp(5)+x**3)*ln(9*x**4
*exp(5)-9*x**5)+4*exp(5)-5*x)/((exp(5)-x)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp(5)-9*x**5))**2+((-2*x*
*2-4*x)*exp(5)+2*x**3+4*x**2)*ln(9*x**4*exp(5)-9*x**5)*ln(1/5*ln(9*x**4*exp(5)-9*x**5))+((x**4+4*x**3+4*x**2)*
exp(5)-x**5-4*x**4-4*x**3)*ln(9*x**4*exp(5)-9*x**5)),x)

[Out]

-x/(-x**2 - 2*x + log(log(-9*x**5 + 9*x**4*exp(5))/5))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x + \log \left (5\right ) - \log \left (i \, \pi + 2 \, \log \left (3\right ) + \log \left (x - e^{5}\right ) + 4 \, \log \left (x\right )\right )} \]

[In]

integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*ex
p(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)
*exp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*
x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, algorithm="maxima")

[Out]

x/(x^2 + 2*x + log(5) - log(I*pi + 2*log(3) + log(x - e^5) + 4*log(x)))

Giac [A] (verification not implemented)

none

Time = 1.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\frac {x}{x^{2} + 2 \, x + \log \left (5\right ) - \log \left (\log \left (-9 \, x^{5} + 9 \, x^{4} e^{5}\right )\right )} \]

[In]

integrate(((-exp(5)+x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+(-x^2*exp(5)+x^3)*log(9*x^4*ex
p(5)-9*x^5)+4*exp(5)-5*x)/((exp(5)-x)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))^2+((-2*x^2-4*x)
*exp(5)+2*x^3+4*x^2)*log(9*x^4*exp(5)-9*x^5)*log(1/5*log(9*x^4*exp(5)-9*x^5))+((x^4+4*x^3+4*x^2)*exp(5)-x^5-4*
x^4-4*x^3)*log(9*x^4*exp(5)-9*x^5)),x, algorithm="giac")

[Out]

x/(x^2 + 2*x + log(5) - log(log(-9*x^5 + 9*x^4*e^5)))

Mupad [F(-1)]

Timed out. \[ \int \frac {4 e^5-5 x+\left (-e^5 x^2+x^3\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (-e^5+x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )}{\left (-4 x^3-4 x^4-x^5+e^5 \left (4 x^2+4 x^3+x^4\right )\right ) \log \left (9 e^5 x^4-9 x^5\right )+\left (4 x^2+2 x^3+e^5 \left (-4 x-2 x^2\right )\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log \left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )+\left (e^5-x\right ) \log \left (9 e^5 x^4-9 x^5\right ) \log ^2\left (\frac {1}{5} \log \left (9 e^5 x^4-9 x^5\right )\right )} \, dx=\text {Hanged} \]

[In]

int((5*x - 4*exp(5) + log(9*x^4*exp(5) - 9*x^5)*(x^2*exp(5) - x^3) - log(log(9*x^4*exp(5) - 9*x^5)/5)*log(9*x^
4*exp(5) - 9*x^5)*(x - exp(5)))/(log(9*x^4*exp(5) - 9*x^5)*(4*x^3 - exp(5)*(4*x^2 + 4*x^3 + x^4) + 4*x^4 + x^5
) + log(log(9*x^4*exp(5) - 9*x^5)/5)^2*log(9*x^4*exp(5) - 9*x^5)*(x - exp(5)) - log(log(9*x^4*exp(5) - 9*x^5)/
5)*log(9*x^4*exp(5) - 9*x^5)*(4*x^2 - exp(5)*(4*x + 2*x^2) + 2*x^3)),x)

[Out]

\text{Hanged}