Integrand size = 182, antiderivative size = 22 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=\log \left (\frac {48 \log ^2\left (\log \left (x+\frac {1}{x^2-\log (2)}\right )\right )}{x^4}\right ) \]
\[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=\int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx \]
Integrate[(-4*x^2 + 2*x^5 - 4*x^3*Log[2] + 2*x*Log[2]^2 + (-4*x^2 - 4*x^5 + (4 + 8*x^3)*Log[2] - 4*x*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log [2])]*Log[Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]])/((x^3 + x^6 + (-x - 2*x^4)*Log[2] + x^2*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]* Log[Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]]),x]
Integrate[(-4*x^2 + 2*x^5 - 4*x^3*Log[2] + 2*x*Log[2]^2 + (-4*x^2 - 4*x^5 + (4 + 8*x^3)*Log[2] - 4*x*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log [2])]*Log[Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]])/((x^3 + x^6 + (-x - 2*x^4)*Log[2] + x^2*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]* Log[Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]]), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 x^5-4 x^3 \log (2)-4 x^2+\left (-4 x^5+\left (8 x^3+4\right ) \log (2)-4 x^2-4 x \log ^2(2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )+2 x \log ^2(2)}{\left (x^6+\left (-2 x^4-x\right ) \log (2)+x^3+x^2 \log ^2(2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^5-4 x^3 \log (2)-4 x^2+\left (-4 x^5+\left (8 x^3+4\right ) \log (2)-4 x^2-4 x \log ^2(2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )+2 x \log ^2(2)}{x \left (x^5-2 x^3 \log (2)+x^2+x \log ^2(2)-\log (2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {2 x^5-4 x^3 \log (2)-4 x^2+\left (-4 x^5+\left (8 x^3+4\right ) \log (2)-4 x^2-4 x \log ^2(2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )+2 x \log ^2(2)}{x \left (x^2-\log (2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )}-\frac {2 x^5-4 x^3 \log (2)-4 x^2+\left (-4 x^5+\left (8 x^3+4\right ) \log (2)-4 x^2-4 x \log ^2(2)\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )+2 x \log ^2(2)}{\left (x^3-x \log (2)+1\right ) \log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right ) \log \left (\log \left (\frac {-x^3+x \log (2)-1}{\log (2)-x^2}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {1}{\left (\sqrt {\log (2)}-x\right ) \log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right ) \log \left (\log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right )\right )}dx-2 \int \frac {1}{\left (x+\sqrt {\log (2)}\right ) \log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right ) \log \left (\log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right )\right )}dx+6 \int \frac {x^2}{\left (x^3-\log (2) x+1\right ) \log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right ) \log \left (\log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right )\right )}dx+2 \log (2) \int \frac {1}{\left (-x^3+\log (2) x-1\right ) \log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right ) \log \left (\log \left (\frac {x^3-\log (2) x+1}{x^2-\log (2)}\right )\right )}dx+x (-\log (16))+4 x \log (2)-4 \log (x)\) |
Int[(-4*x^2 + 2*x^5 - 4*x^3*Log[2] + 2*x*Log[2]^2 + (-4*x^2 - 4*x^5 + (4 + 8*x^3)*Log[2] - 4*x*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]* Log[Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]])/((x^3 + x^6 + (-x - 2*x^4 )*Log[2] + x^2*Log[2]^2)*Log[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]*Log[Lo g[(-1 - x^3 + x*Log[2])/(-x^2 + Log[2])]]),x]
3.24.68.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 16.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(-4 \ln \left (x \right )+2 \ln \left (\ln \left (\ln \left (\frac {x \ln \left (2\right )-x^{3}-1}{\ln \left (2\right )-x^{2}}\right )\right )\right )\) | \(33\) |
| parallelrisch | \(-4 \ln \left (x \right )+2 \ln \left (\ln \left (\ln \left (\frac {x \ln \left (2\right )-x^{3}-1}{\ln \left (2\right )-x^{2}}\right )\right )\right )\) | \(33\) |
| parts | \(-4 \ln \left (x \right )+2 \ln \left (\ln \left (\ln \left (\frac {x \ln \left (2\right )-x^{3}-1}{\ln \left (2\right )-x^{2}}\right )\right )\right )\) | \(33\) |
int(((-4*x*ln(2)^2+(8*x^3+4)*ln(2)-4*x^5-4*x^2)*ln((x*ln(2)-x^3-1)/(ln(2)- x^2))*ln(ln((x*ln(2)-x^3-1)/(ln(2)-x^2)))+2*x*ln(2)^2-4*x^3*ln(2)+2*x^5-4* x^2)/(x^2*ln(2)^2+(-2*x^4-x)*ln(2)+x^6+x^3)/ln((x*ln(2)-x^3-1)/(ln(2)-x^2) )/ln(ln((x*ln(2)-x^3-1)/(ln(2)-x^2))),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=-4 \, \log \left (x\right ) + 2 \, \log \left (\log \left (\log \left (\frac {x^{3} - x \log \left (2\right ) + 1}{x^{2} - \log \left (2\right )}\right )\right )\right ) \]
integrate(((-4*x*log(2)^2+(8*x^3+4)*log(2)-4*x^5-4*x^2)*log((x*log(2)-x^3- 1)/(log(2)-x^2))*log(log((x*log(2)-x^3-1)/(log(2)-x^2)))+2*x*log(2)^2-4*x^ 3*log(2)+2*x^5-4*x^2)/(x^2*log(2)^2+(-2*x^4-x)*log(2)+x^6+x^3)/log((x*log( 2)-x^3-1)/(log(2)-x^2))/log(log((x*log(2)-x^3-1)/(log(2)-x^2))),x, algorit hm=\
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=- 4 \log {\left (x \right )} + 2 \log {\left (\log {\left (\log {\left (\frac {- x^{3} + x \log {\left (2 \right )} - 1}{- x^{2} + \log {\left (2 \right )}} \right )} \right )} \right )} \]
integrate(((-4*x*ln(2)**2+(8*x**3+4)*ln(2)-4*x**5-4*x**2)*ln((x*ln(2)-x**3 -1)/(ln(2)-x**2))*ln(ln((x*ln(2)-x**3-1)/(ln(2)-x**2)))+2*x*ln(2)**2-4*x** 3*ln(2)+2*x**5-4*x**2)/(x**2*ln(2)**2+(-2*x**4-x)*ln(2)+x**6+x**3)/ln((x*l n(2)-x**3-1)/(ln(2)-x**2))/ln(ln((x*ln(2)-x**3-1)/(ln(2)-x**2))),x)
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=-4 \, \log \left (x\right ) + 2 \, \log \left (\log \left (\log \left (x^{3} - x \log \left (2\right ) + 1\right ) - \log \left (x^{2} - \log \left (2\right )\right )\right )\right ) \]
integrate(((-4*x*log(2)^2+(8*x^3+4)*log(2)-4*x^5-4*x^2)*log((x*log(2)-x^3- 1)/(log(2)-x^2))*log(log((x*log(2)-x^3-1)/(log(2)-x^2)))+2*x*log(2)^2-4*x^ 3*log(2)+2*x^5-4*x^2)/(x^2*log(2)^2+(-2*x^4-x)*log(2)+x^6+x^3)/log((x*log( 2)-x^3-1)/(log(2)-x^2))/log(log((x*log(2)-x^3-1)/(log(2)-x^2))),x, algorit hm=\
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=-4 \, \log \left (x\right ) + 2 \, \log \left (\log \left (2 i \, \pi + \log \left (x^{3} - x \log \left (2\right ) + 1\right ) - \log \left (x^{2} - \log \left (2\right )\right )\right )\right ) \]
integrate(((-4*x*log(2)^2+(8*x^3+4)*log(2)-4*x^5-4*x^2)*log((x*log(2)-x^3- 1)/(log(2)-x^2))*log(log((x*log(2)-x^3-1)/(log(2)-x^2)))+2*x*log(2)^2-4*x^ 3*log(2)+2*x^5-4*x^2)/(x^2*log(2)^2+(-2*x^4-x)*log(2)+x^6+x^3)/log((x*log( 2)-x^3-1)/(log(2)-x^2))/log(log((x*log(2)-x^3-1)/(log(2)-x^2))),x, algorit hm=\
Time = 10.83 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-4 x^2+2 x^5-4 x^3 \log (2)+2 x \log ^2(2)+\left (-4 x^2-4 x^5+\left (4+8 x^3\right ) \log (2)-4 x \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )}{\left (x^3+x^6+\left (-x-2 x^4\right ) \log (2)+x^2 \log ^2(2)\right ) \log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right ) \log \left (\log \left (\frac {-1-x^3+x \log (2)}{-x^2+\log (2)}\right )\right )} \, dx=2\,\ln \left (\ln \left (\ln \left (-\frac {x^3-\ln \left (2\right )\,x+1}{\ln \left (2\right )-x^2}\right )\right )\right )-4\,\ln \left (x\right ) \]
int(-(4*x^3*log(2) - 2*x*log(2)^2 + 4*x^2 - 2*x^5 + log(log(-(x^3 - x*log( 2) + 1)/(log(2) - x^2)))*log(-(x^3 - x*log(2) + 1)/(log(2) - x^2))*(4*x*lo g(2)^2 - log(2)*(8*x^3 + 4) + 4*x^2 + 4*x^5))/(log(log(-(x^3 - x*log(2) + 1)/(log(2) - x^2)))*log(-(x^3 - x*log(2) + 1)/(log(2) - x^2))*(x^2*log(2)^ 2 + x^3 + x^6 - log(2)*(x + 2*x^4))),x)