Integrand size = 79, antiderivative size = 16 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 x^4 \log ^2(-4+x-\log (\log (x))) \]
\[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=\int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx \]
Integrate[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x^4)*Log[x] + 64*x^3*Log[x]*Log[Log[x]])*Log[-4 + x - Log[Log[x]]]^2) /((4 - x)*Log[x] + Log[x]*Log[Log[x]]),x]
Integrate[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x^4)*Log[x] + 64*x^3*Log[x]*Log[Log[x]])*Log[-4 + x - Log[Log[x]]]^2) /((4 - x)*Log[x] + Log[x]*Log[Log[x]]), 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.
\(\displaystyle \int \frac {\left (64 x^3 \log (x) \log (\log (x))+\left (256 x^3-64 x^4\right ) \log (x)\right ) \log ^2(x-\log (\log (x))-4)+\left (32 x^3-32 x^4 \log (x)\right ) \log (x-\log (\log (x))-4)}{(4-x) \log (x)+\log (\log (x)) \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {32 x^3 \log (x-\log (\log (x))-4) (-x \log (x)-2 x \log (x-\log (\log (x))-4) \log (x)+2 \log (\log (x)) \log (x-\log (\log (x))-4) \log (x)+8 \log (x-\log (\log (x))-4) \log (x)+1)}{\log (x) (-x+\log (\log (x))+4)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 32 \int \frac {x^3 \log (x-\log (\log (x))-4) (-x \log (x)-2 x \log (x-\log (\log (x))-4) \log (x)+2 \log (\log (x)) \log (x-\log (\log (x))-4) \log (x)+8 \log (x-\log (\log (x))-4) \log (x)+1)}{\log (x) (-x+\log (\log (x))+4)}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle 32 \int \frac {x^3 \log (x-\log (\log (x))-4) (1-\log (x) (x+2 (x-\log (\log (x))-4) \log (x-\log (\log (x))-4)))}{\log (x) (-x+\log (\log (x))+4)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 32 \int \left (2 \log ^2(x-\log (\log (x))-4) x^3+\frac {(x \log (x)-1) \log (x-\log (\log (x))-4) x^3}{\log (x) (x-\log (\log (x))-4)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 32 \left (\int \frac {x^4 \log (x-\log (\log (x))-4)}{x-\log (\log (x))-4}dx+2 \int x^3 \log ^2(x-\log (\log (x))-4)dx-\int \frac {x^3 \log (x-\log (\log (x))-4)}{\log (x) (x-\log (\log (x))-4)}dx\right )\) |
Int[((32*x^3 - 32*x^4*Log[x])*Log[-4 + x - Log[Log[x]]] + ((256*x^3 - 64*x ^4)*Log[x] + 64*x^3*Log[x]*Log[Log[x]])*Log[-4 + x - Log[Log[x]]]^2)/((4 - x)*Log[x] + Log[x]*Log[Log[x]]),x]
3.23.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.88 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
risch | \(16 x^{4} \ln \left (-\ln \left (\ln \left (x \right )\right )+x -4\right )^{2}\) | \(17\) |
parallelrisch | \(16 x^{4} \ln \left (-\ln \left (\ln \left (x \right )\right )+x -4\right )^{2}\) | \(17\) |
int(((64*x^3*ln(x)*ln(ln(x))+(-64*x^4+256*x^3)*ln(x))*ln(-ln(ln(x))+x-4)^2 +(-32*x^4*ln(x)+32*x^3)*ln(-ln(ln(x))+x-4))/(ln(x)*ln(ln(x))+(-x+4)*ln(x)) ,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]
integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(l og(x))+x-4)^2+(-32*x^4*log(x)+32*x^3)*log(-log(log(x))+x-4))/(log(x)*log(l og(x))+(-x+4)*log(x)),x, algorithm=\
Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 x^{4} \log {\left (x - \log {\left (\log {\left (x \right )} \right )} - 4 \right )}^{2} \]
integrate(((64*x**3*ln(x)*ln(ln(x))+(-64*x**4+256*x**3)*ln(x))*ln(-ln(ln(x ))+x-4)**2+(-32*x**4*ln(x)+32*x**3)*ln(-ln(ln(x))+x-4))/(ln(x)*ln(ln(x))+( -x+4)*ln(x)),x)
Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]
integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(l og(x))+x-4)^2+(-32*x^4*log(x)+32*x^3)*log(-log(log(x))+x-4))/(log(x)*log(l og(x))+(-x+4)*log(x)),x, algorithm=\
Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]
integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(l og(x))+x-4)^2+(-32*x^4*log(x)+32*x^3)*log(-log(log(x))+x-4))/(log(x)*log(l og(x))+(-x+4)*log(x)),x, algorithm=\
Time = 14.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16\,x^4\,{\ln \left (x-\ln \left (\ln \left (x\right )\right )-4\right )}^2 \]