\(\int \frac {(-18 x+2 x^3) \log (5)+(-18-6 x+(-6 x^2-2 x^3) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)})+(54-18 x+(18 x^2-6 x^3) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)}) \log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2})+(9-3 x+(3 x^2-x^3) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)}) \log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}) \log (\log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}))}{(-81+81 x-27 x^2+3 x^3+(-27 x^2+27 x^3-9 x^4+x^5) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)}) \log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2})+(-54+36 x-6 x^2+(-18 x^2+12 x^3-2 x^4) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)}) \log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}) \log (\log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}))+(-9+3 x+(-3 x^2+x^3) \log (5)) \log (\frac {3+x^2 \log (5)}{\log (5)}) \log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}) \log ^2(\log (\frac {\log (\frac {3+x^2 \log (5)}{\log (5)})}{9-6 x+x^2}))} \, dx\) [3188]

Optimal result

Integrand size = 460, antiderivative size = 30 \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\frac {3+x}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx \]

[In]

Int[((-18*x + 2*x^3)*Log[5] + (-18 - 6*x + (-6*x^2 - 2*x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]] + (54 - 18*x
+ (18*x^2 - 6*x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)] + (9
- 3*x + (3*x^2 - x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]*
Log[Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]])/((-81 + 81*x - 27*x^2 + 3*x^3 + (-27*x^2 + 27*x^3 - 9*
x^4 + x^5)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)] + (-54 + 36*
x - 6*x^2 + (-18*x^2 + 12*x^3 - 2*x^4)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(
9 - 6*x + x^2)]*Log[Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]] + (-9 + 3*x + (-3*x^2 + x^3)*Log[5])*Lo
g[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]*Log[Log[Log[(3 + x^2*Log[5])/Log[
5]]/(9 - 6*x + x^2)]]^2),x]

[Out]

(-3*Defer[Int][(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^(-2), x])/(1 + Log[125]) - (9*Log[5]*Defer[
Int][(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^(-2), x])/(1 + Log[125]) + (9*Defer[Int][1/((-3 + x)*
(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - (9*(1 - 3*Log[5])*Defer[Int][1/((
-3 + x)*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - (27*Log[5]*Defer[Int][1/(
(-3 + x)*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - Defer[Int][x/(-3 + x - L
og[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2, x]/(1 + Log[125]) - (3*Log[5]*Defer[Int][x/(-3 + x - Log[Log[Log[x
^2 + 3/Log[5]]/(-3 + x)^2]])^2, x])/(1 + Log[125]) + (9*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Lo
g[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*(1 + Log[125])) - (3*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*(-3
+ x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[125])) + (3*(1 - 3*Log[5])*Defer[In
t][1/((-x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[1
25])) + (((9*I)/2)*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2
), x])/(Sqrt[Log[125]]*(1 + Log[125])) - (((9*I)/2)*(1 - 3*Log[5])*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*(-3 +
x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) - (((27*I)/2)*Log[5]*De
fer[Int][1/((-x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]
]*(1 + Log[125])) - (9*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]]
)^2), x])/(2*(1 + Log[125])) + (3*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(
-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[125])) - (3*(1 - 3*Log[5])*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*(-3 +
x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[125])) + (((9*I)/2)*Defer[Int][1/((x
+ I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])
) - (((9*I)/2)*(1 - 3*Log[5])*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 +
x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) - (((27*I)/2)*Log[5]*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*(-3
+ x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) + 2*Defer[Int][1/(Log
[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x] - (4*Defer[Int][1/
(Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]
) - (12*Log[5]*Defer[Int][1/(Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x
)^2]])^2), x])/(1 + Log[125]) - (12*Defer[Int][1/((-3 + x)*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[L
og[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - (36*Log[5]*Defer[Int][1/((-3 + x)*Log[Log[x^2 +
3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - ((3*I)*Defe
r[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-
3 + x)^2]])^2), x])/Sqrt[Log[125]] + ((3*I)*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[Log[x^2 + 3/Log[5]]/(-3
+ x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) + ((9*I)*L
og[5]*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/
Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) - ((3*I)*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*Lo
g[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/Sqrt[Log[125]] +
((3*I)*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3
/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) + ((9*I)*Log[5]*Defer[Int][1/((x + I*Sqrt[3/Log
[5]])*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Lo
g[125]]*(1 + Log[125])) + (3*Log[25]*Defer[Int][1/(Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3
+ x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) + (Log[25]*Defer[Int][1/(Log[x^2 + 3/L
og[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(Log[5]
*(1 + Log[125])) - (18*Log[5]*Defer[Int][1/((-3 + x)*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(
-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) + (9*Log[25]*Defer[Int][1/((-3 + x)*
Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2)
, x])/(1 + Log[125]) - (9*Log[5]*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log
[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125]) - (3*Log[25]*Defer
[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log
[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[125])) - ((9*I)*Log[5]*Defer[Int][1/((-x + I*Sqrt[3/
Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x
)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) - (((3*I)/2)*Log[25]*Defer[Int][1/((-x + I*Sqrt[3/Log[5]])*Log[
x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x]
)/(Log[5]*Sqrt[Log[125]]*(1 + Log[125])) + (9*Log[5]*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*Log[x^2 + 3/Log[5]]*
Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(1 + Log[125])
+ (3*Log[25]*Defer[Int][1/((x + I*Sqrt[3/Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3
+ x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]])^2), x])/(2*Log[5]*(1 + Log[125])) - ((9*I)*Log[5]*Defer[Int][
1/((x + I*Sqrt[3/Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 +
3/Log[5]]/(-3 + x)^2]])^2), x])/(Sqrt[Log[125]]*(1 + Log[125])) - (((3*I)/2)*Log[25]*Defer[Int][1/((x + I*Sqrt
[3/Log[5]])*Log[x^2 + 3/Log[5]]*Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3
+ x)^2]])^2), x])/(Log[5]*Sqrt[Log[125]]*(1 + Log[125])) + Defer[Int][(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-
3 + x)^2]])^(-1), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {18 x \log (5)-x^3 \log (25)+\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2 (3+x)+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (6+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx \\ & = \int \left (\frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}+\frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}\right ) \, dx \\ & = \int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx \\ & = \int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx \\ & = \int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \left (\frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{3 (3-x) (1+\log (125)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}+\frac {(3+x) \log (5) \left (18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{3 \left (3+x^2 \log (5)\right ) (1+\log (125)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}\right ) \, dx \\ & = \frac {\int \frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\frac {\log (5) \int \frac {(3+x) \left (18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx \\ & = \frac {\int \frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{(3-x) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\frac {\log (5) \int \frac {(3+x) \left (18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )}{\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(142\) vs. \(2(30)=60\).

Time = 0.19 (sec) , antiderivative size = 142, normalized size of antiderivative = 4.73 \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{\left (-2 (-3+x) x \log (5)+\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right ) \left (-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )} \]

[In]

Integrate[((-18*x + 2*x^3)*Log[5] + (-18 - 6*x + (-6*x^2 - 2*x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]] + (54 -
18*x + (18*x^2 - 6*x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)
] + (9 - 3*x + (3*x^2 - x^3)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x +
x^2)]*Log[Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]])/((-81 + 81*x - 27*x^2 + 3*x^3 + (-27*x^2 + 27*x^
3 - 9*x^4 + x^5)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)] + (-54
+ 36*x - 6*x^2 + (-18*x^2 + 12*x^3 - 2*x^4)*Log[5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log
[5]]/(9 - 6*x + x^2)]*Log[Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]] + (-9 + 3*x + (-3*x^2 + x^3)*Log[
5])*Log[(3 + x^2*Log[5])/Log[5]]*Log[Log[(3 + x^2*Log[5])/Log[5]]/(9 - 6*x + x^2)]*Log[Log[Log[(3 + x^2*Log[5]
)/Log[5]]/(9 - 6*x + x^2)]]^2),x]

[Out]

(18*x*Log[5] - x^3*Log[25] + (3 + x)*(3 + x^2*Log[5])*Log[x^2 + 3/Log[5]]*(2 + (-3 + x)*Log[Log[x^2 + 3/Log[5]
]/(-3 + x)^2]))/((-2*(-3 + x)*x*Log[5] + (3 + x^2*Log[5])*Log[x^2 + 3/Log[5]]*(2 + (-3 + x)*Log[Log[x^2 + 3/Lo
g[5]]/(-3 + x)^2]))*(-3 + x - Log[Log[Log[x^2 + 3/Log[5]]/(-3 + x)^2]]))

Maple [C] (warning: unable to verify)

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

Time = 0.18 (sec) , antiderivative size = 175, normalized size of antiderivative = 5.83

\[\frac {3+x}{-\ln \left (\ln \left (\ln \left (\frac {x^{2} \ln \left (5\right )+3}{\ln \left (5\right )}\right )\right )-2 \ln \left (-3+x \right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (-3+x \right )^{2}\right )+\operatorname {csgn}\left (i \left (-3+x \right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \left (5\right )+3}{\ln \left (5\right )}\right )}{\left (-3+x \right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \left (5\right )+3}{\ln \left (5\right )}\right )}{\left (-3+x \right )^{2}}\right )+\operatorname {csgn}\left (i \ln \left (\frac {x^{2} \ln \left (5\right )+3}{\ln \left (5\right )}\right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \left (5\right )+3}{\ln \left (5\right )}\right )}{\left (-3+x \right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (-3+x \right )^{2}}\right )\right )}{2}\right )-3+x}\]

[In]

int((((-x^3+3*x^2)*ln(5)-3*x+9)*ln((x^2*ln(5)+3)/ln(5))*ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*x+9))*ln(ln(ln((x^2*
ln(5)+3)/ln(5))/(x^2-6*x+9)))+((-6*x^3+18*x^2)*ln(5)-18*x+54)*ln((x^2*ln(5)+3)/ln(5))*ln(ln((x^2*ln(5)+3)/ln(5
))/(x^2-6*x+9))+((-2*x^3-6*x^2)*ln(5)-6*x-18)*ln((x^2*ln(5)+3)/ln(5))+(2*x^3-18*x)*ln(5))/(((x^3-3*x^2)*ln(5)+
3*x-9)*ln((x^2*ln(5)+3)/ln(5))*ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*x+9))*ln(ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*x+
9)))^2+((-2*x^4+12*x^3-18*x^2)*ln(5)-6*x^2+36*x-54)*ln((x^2*ln(5)+3)/ln(5))*ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*
x+9))*ln(ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*x+9)))+((x^5-9*x^4+27*x^3-27*x^2)*ln(5)+3*x^3-27*x^2+81*x-81)*ln((x
^2*ln(5)+3)/ln(5))*ln(ln((x^2*ln(5)+3)/ln(5))/(x^2-6*x+9))),x)

[Out]

(3+x)/(-ln(ln(ln((x^2*ln(5)+3)/ln(5)))-2*ln(-3+x)+1/2*I*Pi*csgn(I*(-3+x)^2)*(-csgn(I*(-3+x)^2)+csgn(I*(-3+x)))
^2-1/2*I*Pi*csgn(I*ln((x^2*ln(5)+3)/ln(5))/(-3+x)^2)*(-csgn(I*ln((x^2*ln(5)+3)/ln(5))/(-3+x)^2)+csgn(I*ln((x^2
*ln(5)+3)/ln(5))))*(-csgn(I*ln((x^2*ln(5)+3)/ln(5))/(-3+x)^2)+csgn(I/(-3+x)^2)))-3+x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\frac {x + 3}{x - \log \left (\log \left (\frac {\log \left (\frac {x^{2} \log \left (5\right ) + 3}{\log \left (5\right )}\right )}{x^{2} - 6 \, x + 9}\right )\right ) - 3} \]

[In]

integrate((((-x^3+3*x^2)*log(5)-3*x+9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*
log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((-6*x^3+18*x^2)*log(5)-18*x+54)*log((x^2*log(5)+3)/log(5))*l
og(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))+((-2*x^3-6*x^2)*log(5)-6*x-18)*log((x^2*log(5)+3)/log(5))+(2*x^3-18
*x)*log(5))/(((x^3-3*x^2)*log(5)+3*x-9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))
*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))^2+((-2*x^4+12*x^3-18*x^2)*log(5)-6*x^2+36*x-54)*log((x^2*log
(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((
x^5-9*x^4+27*x^3-27*x^2)*log(5)+3*x^3-27*x^2+81*x-81)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5)
)/(x^2-6*x+9))),x, algorithm="fricas")

[Out]

(x + 3)/(x - log(log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9))) - 3)

Sympy [A] (verification not implemented)

Time = 2.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\frac {- x - 3}{- x + \log {\left (\log {\left (\frac {\log {\left (\frac {x^{2} \log {\left (5 \right )} + 3}{\log {\left (5 \right )}} \right )}}{x^{2} - 6 x + 9} \right )} \right )} + 3} \]

[In]

integrate((((-x**3+3*x**2)*ln(5)-3*x+9)*ln((x**2*ln(5)+3)/ln(5))*ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9))*ln(
ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9)))+((-6*x**3+18*x**2)*ln(5)-18*x+54)*ln((x**2*ln(5)+3)/ln(5))*ln(ln((x
**2*ln(5)+3)/ln(5))/(x**2-6*x+9))+((-2*x**3-6*x**2)*ln(5)-6*x-18)*ln((x**2*ln(5)+3)/ln(5))+(2*x**3-18*x)*ln(5)
)/(((x**3-3*x**2)*ln(5)+3*x-9)*ln((x**2*ln(5)+3)/ln(5))*ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9))*ln(ln(ln((x*
*2*ln(5)+3)/ln(5))/(x**2-6*x+9)))**2+((-2*x**4+12*x**3-18*x**2)*ln(5)-6*x**2+36*x-54)*ln((x**2*ln(5)+3)/ln(5))
*ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9))*ln(ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9)))+((x**5-9*x**4+27*x**3
-27*x**2)*ln(5)+3*x**3-27*x**2+81*x-81)*ln((x**2*ln(5)+3)/ln(5))*ln(ln((x**2*ln(5)+3)/ln(5))/(x**2-6*x+9))),x)

[Out]

(-x - 3)/(-x + log(log(log((x**2*log(5) + 3)/log(5))/(x**2 - 6*x + 9))) + 3)

Maxima [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\frac {x + 3}{x - \log \left (-2 \, \log \left (x - 3\right ) + \log \left (\log \left (x^{2} \log \left (5\right ) + 3\right ) - \log \left (\log \left (5\right )\right )\right )\right ) - 3} \]

[In]

integrate((((-x^3+3*x^2)*log(5)-3*x+9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*
log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((-6*x^3+18*x^2)*log(5)-18*x+54)*log((x^2*log(5)+3)/log(5))*l
og(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))+((-2*x^3-6*x^2)*log(5)-6*x-18)*log((x^2*log(5)+3)/log(5))+(2*x^3-18
*x)*log(5))/(((x^3-3*x^2)*log(5)+3*x-9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))
*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))^2+((-2*x^4+12*x^3-18*x^2)*log(5)-6*x^2+36*x-54)*log((x^2*log
(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((
x^5-9*x^4+27*x^3-27*x^2)*log(5)+3*x^3-27*x^2+81*x-81)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5)
)/(x^2-6*x+9))),x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((((-x^3+3*x^2)*log(5)-3*x+9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*
log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((-6*x^3+18*x^2)*log(5)-18*x+54)*log((x^2*log(5)+3)/log(5))*l
og(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))+((-2*x^3-6*x^2)*log(5)-6*x-18)*log((x^2*log(5)+3)/log(5))+(2*x^3-18
*x)*log(5))/(((x^3-3*x^2)*log(5)+3*x-9)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))
*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))^2+((-2*x^4+12*x^3-18*x^2)*log(5)-6*x^2+36*x-54)*log((x^2*log
(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9))*log(log(log((x^2*log(5)+3)/log(5))/(x^2-6*x+9)))+((
x^5-9*x^4+27*x^3-27*x^2)*log(5)+3*x^3-27*x^2+81*x-81)*log((x^2*log(5)+3)/log(5))*log(log((x^2*log(5)+3)/log(5)
)/(x^2-6*x+9))),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error index.cc index_gcd Error: Bad Argument ValueError index.cc index_gcd Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx=\int \frac {\ln \left (5\right )\,\left (18\,x-2\,x^3\right )+\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\left (6\,x+\ln \left (5\right )\,\left (2\,x^3+6\,x^2\right )+18\right )-\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\,\left (\ln \left (5\right )\,\left (18\,x^2-6\,x^3\right )-18\,x+54\right )-\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\right )\,\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\,\left (\ln \left (5\right )\,\left (3\,x^2-x^3\right )-3\,x+9\right )}{\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\,\left (\ln \left (5\right )\,\left (3\,x^2-x^3\right )-3\,x+9\right )\,{\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\right )}^2+\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\,\left (\ln \left (5\right )\,\left (2\,x^4-12\,x^3+18\,x^2\right )-36\,x+6\,x^2+54\right )\,\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\right )+\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \left (5\right )\,x^2+3}{\ln \left (5\right )}\right )}{x^2-6\,x+9}\right )\,\left (\ln \left (5\right )\,\left (-x^5+9\,x^4-27\,x^3+27\,x^2\right )-81\,x+27\,x^2-3\,x^3+81\right )} \,d x \]

[In]

int((log(5)*(18*x - 2*x^3) + log((x^2*log(5) + 3)/log(5))*(6*x + log(5)*(6*x^2 + 2*x^3) + 18) - log((x^2*log(5
) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9))*(log(5)*(18*x^2 - 6*x^3) - 18*x + 54) - log(l
og(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))*log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5)
)/(x^2 - 6*x + 9))*(log(5)*(3*x^2 - x^3) - 3*x + 9))/(log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/lo
g(5))/(x^2 - 6*x + 9))*(log(5)*(27*x^2 - 27*x^3 + 9*x^4 - x^5) - 81*x + 27*x^2 - 3*x^3 + 81) + log(log(log((x^
2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))^2*log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 -
6*x + 9))*(log(5)*(3*x^2 - x^3) - 3*x + 9) + log(log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))*log((x^2*
log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9))*(log(5)*(18*x^2 - 12*x^3 + 2*x^4) - 36*x
+ 6*x^2 + 54)),x)

[Out]

int((log(5)*(18*x - 2*x^3) + log((x^2*log(5) + 3)/log(5))*(6*x + log(5)*(6*x^2 + 2*x^3) + 18) - log((x^2*log(5
) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9))*(log(5)*(18*x^2 - 6*x^3) - 18*x + 54) - log(l
og(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))*log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5)
)/(x^2 - 6*x + 9))*(log(5)*(3*x^2 - x^3) - 3*x + 9))/(log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/lo
g(5))/(x^2 - 6*x + 9))*(log(5)*(27*x^2 - 27*x^3 + 9*x^4 - x^5) - 81*x + 27*x^2 - 3*x^3 + 81) + log(log(log((x^
2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))^2*log((x^2*log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 -
6*x + 9))*(log(5)*(3*x^2 - x^3) - 3*x + 9) + log(log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9)))*log((x^2*
log(5) + 3)/log(5))*log(log((x^2*log(5) + 3)/log(5))/(x^2 - 6*x + 9))*(log(5)*(18*x^2 - 12*x^3 + 2*x^4) - 36*x
+ 6*x^2 + 54)), x)