\(\int \frac {2 x+4 x^2+2 x^3+(2 x+6 x^2+4 x^3) \log (x) \log (\log (x)) \log (\log ^2(\log (x)))}{-e^{25} \log (x) \log (\log (x))+(x^2+2 x^3+x^4) \log (x) \log (\log (x)) \log (\log ^2(\log (x)))} \, dx\) [8670]

Optimal result

Integrand size = 78, antiderivative size = 20 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (e^{25}-\left (x+x^2\right )^2 \log \left (\log ^2(\log (x))\right )\right ) \]

[Out]

ln(exp(25)-(x^2+x)^2*ln(ln(ln(x))^2))

Rubi [F]

\[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx \]

[In]

Int[(2*x + 4*x^2 + 2*x^3 + (2*x + 6*x^2 + 4*x^3)*Log[x]*Log[Log[x]]*Log[Log[Log[x]]^2])/(-(E^25*Log[x]*Log[Log
[x]]) + (x^2 + 2*x^3 + x^4)*Log[x]*Log[Log[x]]*Log[Log[Log[x]]^2]),x]

[Out]

2*Log[x] + 2*Log[1 + x] + 4*E^25*Defer[Int][1/((-1 - x)*(E^25 - x^2*(1 + x)^2*Log[Log[Log[x]]^2])), x] + 2*E^2
5*Defer[Int][1/((1 + x)*(E^25 - x^2*(1 + x)^2*Log[Log[Log[x]]^2])), x] + 2*Defer[Int][x/(Log[x]*Log[Log[x]]*(-
E^25 + x^2*(1 + x)^2*Log[Log[Log[x]]^2])), x] + 4*Defer[Int][x^2/(Log[x]*Log[Log[x]]*(-E^25 + x^2*(1 + x)^2*Lo
g[Log[Log[x]]^2])), x] + 2*Defer[Int][x^3/(Log[x]*Log[Log[x]]*(-E^25 + x^2*(1 + x)^2*Log[Log[Log[x]]^2])), x]
- 2*E^25*Defer[Int][(E^25*x - x^3*(1 + x)^2*Log[Log[Log[x]]^2])^(-1), x]

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (e^{25}-x^2 \log \left (\log ^2(\log (x))\right )-2 x^3 \log \left (\log ^2(\log (x))\right )-x^4 \log \left (\log ^2(\log (x))\right )\right ) \]

[In]

Integrate[(2*x + 4*x^2 + 2*x^3 + (2*x + 6*x^2 + 4*x^3)*Log[x]*Log[Log[x]]*Log[Log[Log[x]]^2])/(-(E^25*Log[x]*L
og[Log[x]]) + (x^2 + 2*x^3 + x^4)*Log[x]*Log[Log[x]]*Log[Log[Log[x]]^2]),x]

[Out]

Log[E^25 - x^2*Log[Log[Log[x]]^2] - 2*x^3*Log[Log[Log[x]]^2] - x^4*Log[Log[Log[x]]^2]]

Maple [A] (verified)

Time = 76.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90

method	result	size
parallelrisch	\(\ln \left (\ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right ) x^{4}+2 \ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right ) x^{3}+x^{2} \ln \left (\ln \left (\ln \left (x \right )\right )^{2}\right )-{\mathrm e}^{25}\right )\)	\(38\)
risch	\(2 \ln \left (x^{2}+x \right )+\ln \left (\ln \left (\ln \left (\ln \left (x \right )\right )\right )-\frac {i \left (\pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-2 \pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+\pi \,x^{4} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}+2 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-4 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+2 \pi \,x^{3} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}+\pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right )^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )-2 \pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{2}+\pi \,x^{2} \operatorname {csgn}\left (i \ln \left (\ln \left (x \right )\right )^{2}\right )^{3}-2 i {\mathrm e}^{25}\right )}{4 x^{2} \left (x^{2}+2 x +1\right )}\right )\)	\(229\)

[In]

int(((4*x^3+6*x^2+2*x)*ln(x)*ln(ln(x))*ln(ln(ln(x))^2)+2*x^3+4*x^2+2*x)/((x^4+2*x^3+x^2)*ln(x)*ln(ln(x))*ln(ln
(ln(x))^2)-exp(25)*ln(x)*ln(ln(x))),x,method=_RETURNVERBOSE)

[Out]

ln(ln(ln(ln(x))^2)*x^4+2*ln(ln(ln(x))^2)*x^3+x^2*ln(ln(ln(x))^2)-exp(25))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \, \log \left (x^{2} + x\right ) + \log \left (\frac {{\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - e^{25}}{x^{4} + 2 \, x^{3} + x^{2}}\right ) \]

[In]

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*lo
g(log(x))*log(log(log(x))^2)-exp(25)*log(x)*log(log(x))),x, algorithm="fricas")

[Out]

2*log(x^2 + x) + log(((x^4 + 2*x^3 + x^2)*log(log(log(x))^2) - e^25)/(x^4 + 2*x^3 + x^2))

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.60 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \log {\left (x^{2} + x \right )} + \log {\left (\log {\left (\log {\left (\log {\left (x \right )} \right )}^{2} \right )} - \frac {e^{25}}{x^{4} + 2 x^{3} + x^{2}} \right )} \]

[In]

integrate(((4*x**3+6*x**2+2*x)*ln(x)*ln(ln(x))*ln(ln(ln(x))**2)+2*x**3+4*x**2+2*x)/((x**4+2*x**3+x**2)*ln(x)*l
n(ln(x))*ln(ln(ln(x))**2)-exp(25)*ln(x)*ln(ln(x))),x)

[Out]

2*log(x**2 + x) + log(log(log(log(x))**2) - exp(25)/(x**4 + 2*x**3 + x**2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).

Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=2 \, \log \left (x + 1\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (\log \left (\log \left (x\right )\right )\right ) - e^{25}}{2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}\right ) \]

[In]

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*lo
g(log(x))*log(log(log(x))^2)-exp(25)*log(x)*log(log(x))),x, algorithm="maxima")

[Out]

2*log(x + 1) + 2*log(x) + log(1/2*(2*(x^4 + 2*x^3 + x^2)*log(log(log(x))) - e^25)/(x^4 + 2*x^3 + x^2))

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=\log \left (-x^{4} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - 2 \, x^{3} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) - x^{2} \log \left (\log \left (\log \left (x\right )\right )^{2}\right ) + e^{25}\right ) \]

[In]

integrate(((4*x^3+6*x^2+2*x)*log(x)*log(log(x))*log(log(log(x))^2)+2*x^3+4*x^2+2*x)/((x^4+2*x^3+x^2)*log(x)*lo
g(log(x))*log(log(log(x))^2)-exp(25)*log(x)*log(log(x))),x, algorithm="giac")

[Out]

log(-x^4*log(log(log(x))^2) - 2*x^3*log(log(log(x))^2) - x^2*log(log(log(x))^2) + e^25)

Mupad [F(-1)]

Timed out. \[ \int \frac {2 x+4 x^2+2 x^3+\left (2 x+6 x^2+4 x^3\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )}{-e^{25} \log (x) \log (\log (x))+\left (x^2+2 x^3+x^4\right ) \log (x) \log (\log (x)) \log \left (\log ^2(\log (x))\right )} \, dx=-\int \frac {2\,x+4\,x^2+2\,x^3+\ln \left ({\ln \left (\ln \left (x\right )\right )}^2\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (4\,x^3+6\,x^2+2\,x\right )}{\ln \left (\ln \left (x\right )\right )\,{\mathrm {e}}^{25}\,\ln \left (x\right )-\ln \left ({\ln \left (\ln \left (x\right )\right )}^2\right )\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (x^4+2\,x^3+x^2\right )} \,d x \]

[In]

int(-(2*x + 4*x^2 + 2*x^3 + log(log(log(x))^2)*log(log(x))*log(x)*(2*x + 6*x^2 + 4*x^3))/(log(log(x))*exp(25)*
log(x) - log(log(log(x))^2)*log(log(x))*log(x)*(x^2 + 2*x^3 + x^4)),x)

[Out]

-int((2*x + 4*x^2 + 2*x^3 + log(log(log(x))^2)*log(log(x))*log(x)*(2*x + 6*x^2 + 4*x^3))/(log(log(x))*exp(25)*
log(x) - log(log(log(x))^2)*log(log(x))*log(x)*(x^2 + 2*x^3 + x^4)), x)