\(\int \frac {(32 x^3-32 x^4 \log (x)) \log (-4+x-\log (\log (x)))+((256 x^3-64 x^4) \log (x)+64 x^3 \log (x) \log (\log (x))) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx\) [8280]

Optimal result

Integrand size = 79, antiderivative size = 16 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 x^4 \log ^2(-4+x-\log (\log (x))) \]

[Out]

16*x^4*ln(-ln(ln(x))+x-4)^2

Rubi [F]

\[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=\int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx \]

[In]

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]

[Out]

32*Defer[Int][(x^4*Log[-4 + x - Log[Log[x]]])/(-4 + x - Log[Log[x]]), x] - 32*Defer[Int][(x^3*Log[-4 + x - Log
[Log[x]]])/(Log[x]*(-4 + x - Log[Log[x]])), x] + 64*Defer[Int][x^3*Log[-4 + x - Log[Log[x]]]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {32 x^3 \log (-4+x-\log (\log (x))) (1-\log (x) (x+2 (-4+x-\log (\log (x))) \log (-4+x-\log (\log (x)))))}{\log (x) (4-x+\log (\log (x)))} \, dx \\ & = 32 \int \frac {x^3 \log (-4+x-\log (\log (x))) (1-\log (x) (x+2 (-4+x-\log (\log (x))) \log (-4+x-\log (\log (x)))))}{\log (x) (4-x+\log (\log (x)))} \, dx \\ & = 32 \int \left (\frac {x^3 (-1+x \log (x)) \log (-4+x-\log (\log (x)))}{\log (x) (-4+x-\log (\log (x)))}+2 x^3 \log ^2(-4+x-\log (\log (x)))\right ) \, dx \\ & = 32 \int \frac {x^3 (-1+x \log (x)) \log (-4+x-\log (\log (x)))}{\log (x) (-4+x-\log (\log (x)))} \, dx+64 \int x^3 \log ^2(-4+x-\log (\log (x))) \, dx \\ & = 32 \int \left (\frac {x^4 \log (-4+x-\log (\log (x)))}{-4+x-\log (\log (x))}-\frac {x^3 \log (-4+x-\log (\log (x)))}{\log (x) (-4+x-\log (\log (x)))}\right ) \, dx+64 \int x^3 \log ^2(-4+x-\log (\log (x))) \, dx \\ & = 32 \int \frac {x^4 \log (-4+x-\log (\log (x)))}{-4+x-\log (\log (x))} \, dx-32 \int \frac {x^3 \log (-4+x-\log (\log (x)))}{\log (x) (-4+x-\log (\log (x)))} \, dx+64 \int x^3 \log ^2(-4+x-\log (\log (x))) \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=\int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx \]

[In]

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]

[Out]

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]

Maple [A] (verified)

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	\(16 x^{4} \ln \left (-\ln \left (\ln \left (x \right )\right )+x -4\right )^{2}\)	\(17\)

[In]

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)

[Out]

16*x^4*ln(-ln(ln(x))+x-4)^2

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="fricas")

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 x^{4} \log {\left (x - \log {\left (\log {\left (x \right )} \right )} - 4 \right )}^{2} \]

[In]

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)

[Out]

16*x**4*log(x - log(log(x)) - 4)**2

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="maxima")

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16 \, x^{4} \log \left (x - \log \left (\log \left (x\right )\right ) - 4\right )^{2} \]

[In]

integrate(((64*x^3*log(x)*log(log(x))+(-64*x^4+256*x^3)*log(x))*log(-log(log(x))+x-4)^2+(-32*x^4*log(x)+32*x^3
)*log(-log(log(x))+x-4))/(log(x)*log(log(x))+(-x+4)*log(x)),x, algorithm="giac")

[Out]

16*x^4*log(x - log(log(x)) - 4)^2

Mupad [B] (verification not implemented)

Time = 14.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\left (32 x^3-32 x^4 \log (x)\right ) \log (-4+x-\log (\log (x)))+\left (\left (256 x^3-64 x^4\right ) \log (x)+64 x^3 \log (x) \log (\log (x))\right ) \log ^2(-4+x-\log (\log (x)))}{(4-x) \log (x)+\log (x) \log (\log (x))} \, dx=16\,x^4\,{\ln \left (x-\ln \left (\ln \left (x\right )\right )-4\right )}^2 \]

[In]

int((log(x - log(log(x)) - 4)*(32*x^4*log(x) - 32*x^3) - log(x - log(log(x)) - 4)^2*(log(x)*(256*x^3 - 64*x^4)
+ 64*x^3*log(log(x))*log(x)))/(log(x)*(x - 4) - log(log(x))*log(x)),x)

[Out]

16*x^4*log(x - log(log(x)) - 4)^2