\(\int \frac {6 x^2+6 x^3+(-216+1296 x+54 x^2-396 x^3) \log (x)+(108-648 x-9 x^2+210 x^3) \log (x) \log (\log (x))+(-18+108 x-36 x^3) \log (x) \log ^2(\log (x))+(1-6 x+2 x^3) \log (x) \log ^3(\log (x))}{(-216-432 x-216 x^2) \log (x)+(108+216 x+108 x^2) \log (x) \log (\log (x))+(-18-36 x-18 x^2) \log (x) \log ^2(\log (x))+(1+2 x+x^2) \log (x) \log ^3(\log (x))} \, dx\) [1305]

Optimal result

Integrand size = 153, antiderivative size = 30 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x \left (-4+x-\frac {5-\frac {3 x^2}{(6-\log (\log (x)))^2}}{-1-x}\right ) \]

[Out]

x*(x-4-(5-3*x^2/(6-ln(ln(x)))^2)/(-1-x))

Rubi [F]

\[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx \]

[In]

Int[(6*x^2 + 6*x^3 + (-216 + 1296*x + 54*x^2 - 396*x^3)*Log[x] + (108 - 648*x - 9*x^2 + 210*x^3)*Log[x]*Log[Lo
g[x]] + (-18 + 108*x - 36*x^3)*Log[x]*Log[Log[x]]^2 + (1 - 6*x + 2*x^3)*Log[x]*Log[Log[x]]^3)/((-216 - 432*x -
216*x^2)*Log[x] + (108 + 216*x + 108*x^2)*Log[x]*Log[Log[x]] + (-18 - 36*x - 18*x^2)*Log[x]*Log[Log[x]]^2 + (
1 + 2*x + x^2)*Log[x]*Log[Log[x]]^3),x]

[Out]

-4*x + x^2 - 5/(1 + x) - 6*Defer[Int][1/(Log[x]*(-6 + Log[Log[x]])^3), x] + 6*Defer[Int][x/(Log[x]*(-6 + Log[L
og[x]])^3), x] + 6*Defer[Int][1/((1 + x)*Log[x]*(-6 + Log[Log[x]])^3), x] + 3*Defer[Int][(-6 + Log[Log[x]])^(-
2), x] - 6*Defer[Int][x/(-6 + Log[Log[x]])^2, x] - 3*Defer[Int][1/((1 + x)^2*(-6 + Log[Log[x]])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-6 x^2 (1+x)-\log (x) (-6+\log (\log (x))) \left (36-216 x-9 x^2+66 x^3-12 \left (1-6 x+2 x^3\right ) \log (\log (x))+\left (1-6 x+2 x^3\right ) \log ^2(\log (x))\right )}{(1+x)^2 \log (x) (6-\log (\log (x)))^3} \, dx \\ & = \int \left (\frac {1-6 x+2 x^3}{(1+x)^2}+\frac {6 x^2}{(1+x) \log (x) (-6+\log (\log (x)))^3}-\frac {3 x^2 (3+2 x)}{(1+x)^2 (-6+\log (\log (x)))^2}\right ) \, dx \\ & = -\left (3 \int \frac {x^2 (3+2 x)}{(1+x)^2 (-6+\log (\log (x)))^2} \, dx\right )+6 \int \frac {x^2}{(1+x) \log (x) (-6+\log (\log (x)))^3} \, dx+\int \frac {1-6 x+2 x^3}{(1+x)^2} \, dx \\ & = -\left (3 \int \left (-\frac {1}{(-6+\log (\log (x)))^2}+\frac {2 x}{(-6+\log (\log (x)))^2}+\frac {1}{(1+x)^2 (-6+\log (\log (x)))^2}\right ) \, dx\right )+6 \int \left (-\frac {1}{\log (x) (-6+\log (\log (x)))^3}+\frac {x}{\log (x) (-6+\log (\log (x)))^3}+\frac {1}{(1+x) \log (x) (-6+\log (\log (x)))^3}\right ) \, dx+\int \left (-4+2 x+\frac {5}{(1+x)^2}\right ) \, dx \\ & = -4 x+x^2-\frac {5}{1+x}+3 \int \frac {1}{(-6+\log (\log (x)))^2} \, dx-3 \int \frac {1}{(1+x)^2 (-6+\log (\log (x)))^2} \, dx-6 \int \frac {1}{\log (x) (-6+\log (\log (x)))^3} \, dx+6 \int \frac {x}{\log (x) (-6+\log (\log (x)))^3} \, dx+6 \int \frac {1}{(1+x) \log (x) (-6+\log (\log (x)))^3} \, dx-6 \int \frac {x}{(-6+\log (\log (x)))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=-4 x+x^2-\frac {5}{1+x}-\frac {3 x^3}{(1+x) (-6+\log (\log (x)))^2} \]

[In]

Integrate[(6*x^2 + 6*x^3 + (-216 + 1296*x + 54*x^2 - 396*x^3)*Log[x] + (108 - 648*x - 9*x^2 + 210*x^3)*Log[x]*
Log[Log[x]] + (-18 + 108*x - 36*x^3)*Log[x]*Log[Log[x]]^2 + (1 - 6*x + 2*x^3)*Log[x]*Log[Log[x]]^3)/((-216 - 4
32*x - 216*x^2)*Log[x] + (108 + 216*x + 108*x^2)*Log[x]*Log[Log[x]] + (-18 - 36*x - 18*x^2)*Log[x]*Log[Log[x]]
^2 + (1 + 2*x + x^2)*Log[x]*Log[Log[x]]^3),x]

[Out]

-4*x + x^2 - 5/(1 + x) - (3*x^3)/((1 + x)*(-6 + Log[Log[x]])^2)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27

method	result	size
risch	\(\frac {x^{3}-3 x^{2}-4 x -5}{1+x}-\frac {3 x^{3}}{\left (1+x \right ) \left (\ln \left (\ln \left (x \right )\right )-6\right )^{2}}\)	\(38\)
parallelrisch	\(-\frac {432+144 \ln \left (\ln \left (x \right )\right ) x^{3}+36 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-432 x^{2} \ln \left (\ln \left (x \right )\right )+12 \ln \left (\ln \left (x \right )\right )^{2}-144 \ln \left (\ln \left (x \right )\right )-396 x^{3}+1296 x^{2}-12 \ln \left (\ln \left (x \right )\right )^{2} x^{3}}{12 \left (\ln \left (\ln \left (x \right )\right )^{2}-12 \ln \left (\ln \left (x \right )\right )+36\right ) \left (1+x \right )}\)	\(82\)

[In]

int(((2*x^3-6*x+1)*ln(x)*ln(ln(x))^3+(-36*x^3+108*x-18)*ln(x)*ln(ln(x))^2+(210*x^3-9*x^2-648*x+108)*ln(x)*ln(l
n(x))+(-396*x^3+54*x^2+1296*x-216)*ln(x)+6*x^3+6*x^2)/((x^2+2*x+1)*ln(x)*ln(ln(x))^3+(-18*x^2-36*x-18)*ln(x)*l
n(ln(x))^2+(108*x^2+216*x+108)*ln(x)*ln(ln(x))+(-216*x^2-432*x-216)*ln(x)),x,method=_RETURNVERBOSE)

[Out]

(x^3-3*x^2-4*x-5)/(1+x)-3*x^3/(1+x)/(ln(ln(x))-6)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {33 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} - 108 \, x^{2} - 12 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right ) - 144 \, x - 180}{{\left (x + 1\right )} \log \left (\log \left (x\right )\right )^{2} - 12 \, {\left (x + 1\right )} \log \left (\log \left (x\right )\right ) + 36 \, x + 36} \]

[In]

integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*log(log(x))^2+(210*x^3-9*x^2-648*x+108
)*log(x)*log(log(x))+(-396*x^3+54*x^2+1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x
^2-36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-216*x^2-432*x-216)*log(x)),x, algori
thm="fricas")

[Out]

(33*x^3 + (x^3 - 3*x^2 - 4*x - 5)*log(log(x))^2 - 108*x^2 - 12*(x^3 - 3*x^2 - 4*x - 5)*log(log(x)) - 144*x - 1
80)/((x + 1)*log(log(x))^2 - 12*(x + 1)*log(log(x)) + 36*x + 36)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=- \frac {3 x^{3}}{36 x + \left (- 12 x - 12\right ) \log {\left (\log {\left (x \right )} \right )} + \left (x + 1\right ) \log {\left (\log {\left (x \right )} \right )}^{2} + 36} + x^{2} - 4 x - \frac {5}{x + 1} \]

[In]

integrate(((2*x**3-6*x+1)*ln(x)*ln(ln(x))**3+(-36*x**3+108*x-18)*ln(x)*ln(ln(x))**2+(210*x**3-9*x**2-648*x+108
)*ln(x)*ln(ln(x))+(-396*x**3+54*x**2+1296*x-216)*ln(x)+6*x**3+6*x**2)/((x**2+2*x+1)*ln(x)*ln(ln(x))**3+(-18*x*
*2-36*x-18)*ln(x)*ln(ln(x))**2+(108*x**2+216*x+108)*ln(x)*ln(ln(x))+(-216*x**2-432*x-216)*ln(x)),x)

[Out]

-3*x**3/(36*x + (-12*x - 12)*log(log(x)) + (x + 1)*log(log(x))**2 + 36) + x**2 - 4*x - 5/(x + 1)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.57 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=\frac {33 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right )^{2} - 108 \, x^{2} - 12 \, {\left (x^{3} - 3 \, x^{2} - 4 \, x - 5\right )} \log \left (\log \left (x\right )\right ) - 144 \, x - 180}{{\left (x + 1\right )} \log \left (\log \left (x\right )\right )^{2} - 12 \, {\left (x + 1\right )} \log \left (\log \left (x\right )\right ) + 36 \, x + 36} \]

[In]

integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*log(log(x))^2+(210*x^3-9*x^2-648*x+108
)*log(x)*log(log(x))+(-396*x^3+54*x^2+1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x
^2-36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-216*x^2-432*x-216)*log(x)),x, algori
thm="maxima")

[Out]

(33*x^3 + (x^3 - 3*x^2 - 4*x - 5)*log(log(x))^2 - 108*x^2 - 12*(x^3 - 3*x^2 - 4*x - 5)*log(log(x)) - 144*x - 1
80)/((x + 1)*log(log(x))^2 - 12*(x + 1)*log(log(x)) + 36*x + 36)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x^{2} - \frac {3 \, x^{3}}{x \log \left (\log \left (x\right )\right )^{2} - 12 \, x \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} + 36 \, x - 12 \, \log \left (\log \left (x\right )\right ) + 36} - 4 \, x - \frac {5}{x + 1} \]

[In]

integrate(((2*x^3-6*x+1)*log(x)*log(log(x))^3+(-36*x^3+108*x-18)*log(x)*log(log(x))^2+(210*x^3-9*x^2-648*x+108
)*log(x)*log(log(x))+(-396*x^3+54*x^2+1296*x-216)*log(x)+6*x^3+6*x^2)/((x^2+2*x+1)*log(x)*log(log(x))^3+(-18*x
^2-36*x-18)*log(x)*log(log(x))^2+(108*x^2+216*x+108)*log(x)*log(log(x))+(-216*x^2-432*x-216)*log(x)),x, algori
thm="giac")

[Out]

x^2 - 3*x^3/(x*log(log(x))^2 - 12*x*log(log(x)) + log(log(x))^2 + 36*x - 12*log(log(x)) + 36) - 4*x - 5/(x + 1
)

Mupad [B] (verification not implemented)

Time = 8.49 (sec) , antiderivative size = 259, normalized size of antiderivative = 8.63 \[ \int \frac {6 x^2+6 x^3+\left (-216+1296 x+54 x^2-396 x^3\right ) \log (x)+\left (108-648 x-9 x^2+210 x^3\right ) \log (x) \log (\log (x))+\left (-18+108 x-36 x^3\right ) \log (x) \log ^2(\log (x))+\left (1-6 x+2 x^3\right ) \log (x) \log ^3(\log (x))}{\left (-216-432 x-216 x^2\right ) \log (x)+\left (108+216 x+108 x^2\right ) \log (x) \log (\log (x))+\left (-18-36 x-18 x^2\right ) \log (x) \log ^2(\log (x))+\left (1+2 x+x^2\right ) \log (x) \log ^3(\log (x))} \, dx=x^2-\frac {5}{x+1}-\frac {\frac {3\,x\,\left (9\,x^2\,\ln \left (x\right )+6\,x^3\,\ln \left (x\right )+x^2+x^3\right )}{{\left (x+1\right )}^2}-\frac {3\,x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (2\,x^3+3\,x^2\right )}{2\,{\left (x+1\right )}^2}}{{\ln \left (\ln \left (x\right )\right )}^2-12\,\ln \left (\ln \left (x\right )\right )+36}-\frac {\frac {3\,x\,\ln \left (x\right )\,\left (54\,x^2\,\ln \left (x\right )+66\,x^3\,\ln \left (x\right )+24\,x^4\,\ln \left (x\right )+21\,x^2+35\,x^3+14\,x^4\right )}{2\,{\left (x+1\right )}^3}-\frac {3\,x\,\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (9\,x^2\,\ln \left (x\right )+11\,x^3\,\ln \left (x\right )+4\,x^4\,\ln \left (x\right )+3\,x^2+5\,x^3+2\,x^4\right )}{2\,{\left (x+1\right )}^3}}{\ln \left (\ln \left (x\right )\right )-6}-4\,x+{\ln \left (x\right )}^2\,\left (\frac {-6\,x^5-\frac {33\,x^4}{2}+\frac {81\,x^2}{2}+\frac {81\,x}{2}+\frac {27}{2}}{x^3+3\,x^2+3\,x+1}-\frac {27}{2}\right )-\frac {\ln \left (x\right )\,\left (3\,x^4+\frac {9\,x^3}{2}\right )}{x^2+2\,x+1} \]

[In]

int(-(6*x^2 + 6*x^3 + log(x)*(1296*x + 54*x^2 - 396*x^3 - 216) - log(log(x))*log(x)*(648*x + 9*x^2 - 210*x^3 -
108) + log(log(x))^3*log(x)*(2*x^3 - 6*x + 1) - log(log(x))^2*log(x)*(36*x^3 - 108*x + 18))/(log(x)*(432*x +
216*x^2 + 216) - log(log(x))*log(x)*(216*x + 108*x^2 + 108) - log(log(x))^3*log(x)*(2*x + x^2 + 1) + log(log(x
))^2*log(x)*(36*x + 18*x^2 + 18)),x)

[Out]

x^2 - 5/(x + 1) - ((3*x*(9*x^2*log(x) + 6*x^3*log(x) + x^2 + x^3))/(x + 1)^2 - (3*x*log(log(x))*log(x)*(3*x^2
+ 2*x^3))/(2*(x + 1)^2))/(log(log(x))^2 - 12*log(log(x)) + 36) - ((3*x*log(x)*(54*x^2*log(x) + 66*x^3*log(x) +
24*x^4*log(x) + 21*x^2 + 35*x^3 + 14*x^4))/(2*(x + 1)^3) - (3*x*log(log(x))*log(x)*(9*x^2*log(x) + 11*x^3*log
(x) + 4*x^4*log(x) + 3*x^2 + 5*x^3 + 2*x^4))/(2*(x + 1)^3))/(log(log(x)) - 6) - 4*x + log(x)^2*(((81*x)/2 + (8
1*x^2)/2 - (33*x^4)/2 - 6*x^5 + 27/2)/(3*x + 3*x^2 + x^3 + 1) - 27/2) - (log(x)*((9*x^3)/2 + 3*x^4))/(2*x + x^
2 + 1)