\(\int \frac {9+3 e^{e^5 x} x-3 x^2+(-18+18 x^2+e^{e^5 x} (-12 x-6 e^5 x^2)) \log (x) \log (\log (x))}{(27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} (9 x^5-3 x^7)+e^{e^5 x} (27 x^4-18 x^6+3 x^8)) \log (x)} \, dx\) [4781]

Optimal result

Integrand size = 134, antiderivative size = 28 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\log (5)+\frac {3 \log (\log (x))}{x^2 \left (-3+x \left (-e^{e^5 x}+x\right )\right )^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx \]

[In]

Int[(9 + 3*E^(E^5*x)*x - 3*x^2 + (-18 + 18*x^2 + E^(E^5*x)*(-12*x - 6*E^5*x^2))*Log[x]*Log[Log[x]])/((27*x^3 -
27*x^5 + E^(3*E^5*x)*x^6 + 9*x^7 - x^9 + E^(2*E^5*x)*(9*x^5 - 3*x^7) + E^(E^5*x)*(27*x^4 - 18*x^6 + 3*x^8))*L
og[x]),x]

[Out]

3*Defer[Int][1/(x^3*(-3 - E^(E^5*x)*x + x^2)^2*Log[x]), x] - 6*E^5*Defer[Int][Log[Log[x]]/(3 + E^(E^5*x)*x - x
^2)^3, x] - 18*Defer[Int][Log[Log[x]]/(x^3*(-3 - E^(E^5*x)*x + x^2)^3), x] - 18*E^5*Defer[Int][Log[Log[x]]/(x^
2*(-3 - E^(E^5*x)*x + x^2)^3), x] - 6*Defer[Int][Log[Log[x]]/(x*(-3 - E^(E^5*x)*x + x^2)^3), x] - 12*Defer[Int
][Log[Log[x]]/(x^3*(-3 - E^(E^5*x)*x + x^2)^2), x] - 6*E^5*Defer[Int][Log[Log[x]]/(x^2*(-3 - E^(E^5*x)*x + x^2
)^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {3 \left (3+e^{e^5 x} x-x^2-2 \left (3+2 e^{e^5 x} x-3 x^2+e^{5+e^5 x} x^2\right ) \log (x) \log (\log (x))\right )}{x^3 \left (3+e^{e^5 x} x-x^2\right )^3 \log (x)} \, dx \\ & = 3 \int \frac {3+e^{e^5 x} x-x^2-2 \left (3+2 e^{e^5 x} x-3 x^2+e^{5+e^5 x} x^2\right ) \log (x) \log (\log (x))}{x^3 \left (3+e^{e^5 x} x-x^2\right )^3 \log (x)} \, dx \\ & = 3 \int \left (\frac {2 \left (-3-3 e^5 x-x^2+e^5 x^3\right ) \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^3}-\frac {-1+4 \log (x) \log (\log (x))+2 e^5 x \log (x) \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2 \log (x)}\right ) \, dx \\ & = -\left (3 \int \frac {-1+4 \log (x) \log (\log (x))+2 e^5 x \log (x) \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2 \log (x)} \, dx\right )+6 \int \frac {\left (-3-3 e^5 x-x^2+e^5 x^3\right ) \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^3} \, dx \\ & = -\left (3 \int \left (-\frac {1}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2 \log (x)}+\frac {4 \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2}+\frac {2 e^5 \log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^2}\right ) \, dx\right )+6 \int \left (-\frac {e^5 \log (\log (x))}{\left (3+e^{e^5 x} x-x^2\right )^3}-\frac {3 \log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^3}-\frac {3 e^5 \log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^3}-\frac {\log (\log (x))}{x \left (-3-e^{e^5 x} x+x^2\right )^3}\right ) \, dx \\ & = 3 \int \frac {1}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2 \log (x)} \, dx-6 \int \frac {\log (\log (x))}{x \left (-3-e^{e^5 x} x+x^2\right )^3} \, dx-12 \int \frac {\log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^2} \, dx-18 \int \frac {\log (\log (x))}{x^3 \left (-3-e^{e^5 x} x+x^2\right )^3} \, dx-\left (6 e^5\right ) \int \frac {\log (\log (x))}{\left (3+e^{e^5 x} x-x^2\right )^3} \, dx-\left (6 e^5\right ) \int \frac {\log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^2} \, dx-\left (18 e^5\right ) \int \frac {\log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \log (\log (x))}{x^2 \left (-3-e^{e^5 x} x+x^2\right )^2} \]

[In]

Integrate[(9 + 3*E^(E^5*x)*x - 3*x^2 + (-18 + 18*x^2 + E^(E^5*x)*(-12*x - 6*E^5*x^2))*Log[x]*Log[Log[x]])/((27
*x^3 - 27*x^5 + E^(3*E^5*x)*x^6 + 9*x^7 - x^9 + E^(2*E^5*x)*(9*x^5 - 3*x^7) + E^(E^5*x)*(27*x^4 - 18*x^6 + 3*x
^8))*Log[x]),x]

[Out]

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

Maple [A] (verified)

Time = 199.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86

method	result	size
risch	\(\frac {3 \ln \left (\ln \left (x \right )\right )}{x^{2} \left (x^{2}-x \,{\mathrm e}^{x \,{\mathrm e}^{5}}-3\right )^{2}}\)	\(24\)
parallelrisch	\(\frac {3 \ln \left (\ln \left (x \right )\right )}{x^{2} \left (x^{4}-2 \,{\mathrm e}^{x \,{\mathrm e}^{5}} x^{3}+{\mathrm e}^{2 x \,{\mathrm e}^{5}} x^{2}-6 x^{2}+6 x \,{\mathrm e}^{x \,{\mathrm e}^{5}}+9\right )}\)	\(50\)

[In]

int((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*ln(x)*ln(ln(x))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*exp(x*exp(
5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/ln(x),x,meth
od=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{5} - 3 \, x^{3}\right )} e^{\left (x e^{5}\right )}} \]

[In]

integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*e
xp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log
(x),x, algorithm="fricas")

[Out]

3*log(log(x))/(x^6 + x^4*e^(2*x*e^5) - 6*x^4 + 9*x^2 - 2*(x^5 - 3*x^3)*e^(x*e^5))

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \log {\left (\log {\left (x \right )} \right )}}{x^{6} + x^{4} e^{2 x e^{5}} - 6 x^{4} + 9 x^{2} + \left (- 2 x^{5} + 6 x^{3}\right ) e^{x e^{5}}} \]

[In]

integrate((((-6*x**2*exp(5)-12*x)*exp(x*exp(5))+18*x**2-18)*ln(x)*ln(ln(x))+3*x*exp(x*exp(5))-3*x**2+9)/(x**6*
exp(x*exp(5))**3+(-3*x**7+9*x**5)*exp(x*exp(5))**2+(3*x**8-18*x**6+27*x**4)*exp(x*exp(5))-x**9+9*x**7-27*x**5+
27*x**3)/ln(x),x)

[Out]

3*log(log(x))/(x**6 + x**4*exp(2*x*exp(5)) - 6*x**4 + 9*x**2 + (-2*x**5 + 6*x**3)*exp(x*exp(5)))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{5} - 3 \, x^{3}\right )} e^{\left (x e^{5}\right )}} \]

[In]

integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*e
xp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log
(x),x, algorithm="maxima")

[Out]

3*log(log(x))/(x^6 + x^4*e^(2*x*e^5) - 6*x^4 + 9*x^2 - 2*(x^5 - 3*x^3)*e^(x*e^5))

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3 \, \log \left (\log \left (x\right )\right )}{x^{6} - 2 \, x^{5} e^{\left (x e^{5}\right )} + x^{4} e^{\left (2 \, x e^{5}\right )} - 6 \, x^{4} + 6 \, x^{3} e^{\left (x e^{5}\right )} + 9 \, x^{2}} \]

[In]

integrate((((-6*x^2*exp(5)-12*x)*exp(x*exp(5))+18*x^2-18)*log(x)*log(log(x))+3*x*exp(x*exp(5))-3*x^2+9)/(x^6*e
xp(x*exp(5))^3+(-3*x^7+9*x^5)*exp(x*exp(5))^2+(3*x^8-18*x^6+27*x^4)*exp(x*exp(5))-x^9+9*x^7-27*x^5+27*x^3)/log
(x),x, algorithm="giac")

[Out]

3*log(log(x))/(x^6 - 2*x^5*e^(x*e^5) + x^4*e^(2*x*e^5) - 6*x^4 + 6*x^3*e^(x*e^5) + 9*x^2)

Mupad [B] (verification not implemented)

Time = 11.30 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {9+3 e^{e^5 x} x-3 x^2+\left (-18+18 x^2+e^{e^5 x} \left (-12 x-6 e^5 x^2\right )\right ) \log (x) \log (\log (x))}{\left (27 x^3-27 x^5+e^{3 e^5 x} x^6+9 x^7-x^9+e^{2 e^5 x} \left (9 x^5-3 x^7\right )+e^{e^5 x} \left (27 x^4-18 x^6+3 x^8\right )\right ) \log (x)} \, dx=\frac {3\,\ln \left (\ln \left (x\right )\right )}{x^2\,\left ({\mathrm {e}}^{x\,{\mathrm {e}}^5}\,\left (6\,x-2\,x^3\right )-6\,x^2+x^4+x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^5}+9\right )} \]

[In]

int((3*x*exp(x*exp(5)) - 3*x^2 - log(log(x))*log(x)*(exp(x*exp(5))*(12*x + 6*x^2*exp(5)) - 18*x^2 + 18) + 9)/(
log(x)*(exp(x*exp(5))*(27*x^4 - 18*x^6 + 3*x^8) + exp(2*x*exp(5))*(9*x^5 - 3*x^7) + 27*x^3 - 27*x^5 + 9*x^7 -
x^9 + x^6*exp(3*x*exp(5)))),x)

[Out]

(3*log(log(x)))/(x^2*(exp(x*exp(5))*(6*x - 2*x^3) - 6*x^2 + x^4 + x^2*exp(2*x*exp(5)) + 9))