\(\int \frac {-32-80 x+2 x^2-16 x \log (x)}{(1680 x+800 x^2+95 x^3+(656 x+320 x^2+39 x^3) \log (x)+(64 x+32 x^2+4 x^3) \log ^2(x)) \log (\frac {28224+12768 x+1444 x^2+(10752+5120 x+608 x^2) \log (x)+(1024+512 x+64 x^2) \log ^2(x)}{400+200 x+25 x^2+(160+80 x+10 x^2) \log (x)+(16+8 x+x^2) \log ^2(x)})} \, dx\) [2058]

Optimal result

Integrand size = 148, antiderivative size = 24 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log \left (\log \left (4 \left (4-\frac {-4+x}{(4+x) (5+\log (x))}\right )^2\right )\right ) \] Output:

ln(ln(2*(4-(-4+x)/(5+ln(x))/(4+x))*(8-2*(-4+x)/(5+ln(x))/(4+x))))
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 (84+19 x+4 (4+x) \log (x))^2}{(4+x)^2 (5+\log (x))^2}\right )\right ) \] Input:

Integrate[(-32 - 80*x + 2*x^2 - 16*x*Log[x])/((1680*x + 800*x^2 + 95*x^3 + 
 (656*x + 320*x^2 + 39*x^3)*Log[x] + (64*x + 32*x^2 + 4*x^3)*Log[x]^2)*Log 
[(28224 + 12768*x + 1444*x^2 + (10752 + 5120*x + 608*x^2)*Log[x] + (1024 + 
 512*x + 64*x^2)*Log[x]^2)/(400 + 200*x + 25*x^2 + (160 + 80*x + 10*x^2)*L 
og[x] + (16 + 8*x + x^2)*Log[x]^2)]),x]
 

Output:

Log[Log[(4*(84 + 19*x + 4*(4 + x)*Log[x])^2)/((4 + x)^2*(5 + Log[x])^2)]]
 

Rubi [F]

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.

\(\displaystyle \int \frac {2 x^2-80 x-16 x \log (x)-32}{\left (95 x^3+800 x^2+\left (4 x^3+32 x^2+64 x\right ) \log ^2(x)+\left (39 x^3+320 x^2+656 x\right ) \log (x)+1680 x\right ) \log \left (\frac {1444 x^2+\left (64 x^2+512 x+1024\right ) \log ^2(x)+\left (608 x^2+5120 x+10752\right ) \log (x)+12768 x+28224}{25 x^2+\left (x^2+8 x+16\right ) \log ^2(x)+\left (10 x^2+80 x+160\right ) \log (x)+200 x+400}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (x^2-40 x-8 x \log (x)-16\right )}{x (x+4) (\log (x)+5) (19 x+4 (x+4) \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {-x^2+8 \log (x) x+40 x+16}{x (x+4) (\log (x)+5) (19 x+4 (x+4) \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {-x^2+8 \log (x) x+40 x+16}{x (x+4) (\log (x)+5) (19 x+4 (x+4) \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {x^2-8 \log (x) x-40 x-16}{4 (x+4) (\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}+\frac {-x^2+8 \log (x) x+40 x+16}{4 x (\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \left (-\int \frac {1}{(\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx+4 \int \frac {1}{x (\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx+40 \int \frac {1}{(x+4) (\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx+8 \int \frac {\log (x)}{(x+4) (\log (x)+5) (4 \log (x) x+19 x+16 \log (x)+84) \log \left (\frac {4 (19 x+4 (x+4) \log (x)+84)^2}{(x+4)^2 (\log (x)+5)^2}\right )}dx\right )\)

Input:

Int[(-32 - 80*x + 2*x^2 - 16*x*Log[x])/((1680*x + 800*x^2 + 95*x^3 + (656* 
x + 320*x^2 + 39*x^3)*Log[x] + (64*x + 32*x^2 + 4*x^3)*Log[x]^2)*Log[(2822 
4 + 12768*x + 1444*x^2 + (10752 + 5120*x + 608*x^2)*Log[x] + (1024 + 512*x 
 + 64*x^2)*Log[x]^2)/(400 + 200*x + 25*x^2 + (160 + 80*x + 10*x^2)*Log[x] 
+ (16 + 8*x + x^2)*Log[x]^2)]),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(40)=80\).

Time = 6.48 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.79

method	result	size
parallelrisch	\(\ln \left (\ln \left (\frac {\left (64 x^{2}+512 x +1024\right ) \ln \left (x \right )^{2}+\left (608 x^{2}+5120 x +10752\right ) \ln \left (x \right )+1444 x^{2}+12768 x +28224}{x^{2} \ln \left (x \right )^{2}+8 x \ln \left (x \right )^{2}+10 x^{2} \ln \left (x \right )+16 \ln \left (x \right )^{2}+80 x \ln \left (x \right )+25 x^{2}+160 \ln \left (x \right )+200 x +400}\right )\right )\)	\(91\)
default	\(\ln \left (2 \ln \left (2\right )+\ln \left (\frac {16 x^{2} \ln \left (x \right )^{2}+128 x \ln \left (x \right )^{2}+152 x^{2} \ln \left (x \right )+256 \ln \left (x \right )^{2}+1280 x \ln \left (x \right )+361 x^{2}+2688 \ln \left (x \right )+3192 x +7056}{x^{2} \ln \left (x \right )^{2}+8 x \ln \left (x \right )^{2}+10 x^{2} \ln \left (x \right )+16 \ln \left (x \right )^{2}+80 x \ln \left (x \right )+25 x^{2}+160 \ln \left (x \right )+200 x +400}\right )\right )\)	\(106\)
risch	\(\text {Expression too large to display}\)	\(622\)

Input:

int((-16*x*ln(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*ln(x)^2+(39*x^3+320*x 
^2+656*x)*ln(x)+95*x^3+800*x^2+1680*x)/ln(((64*x^2+512*x+1024)*ln(x)^2+(60 
8*x^2+5120*x+10752)*ln(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16)*ln(x)^2+(1 
0*x^2+80*x+160)*ln(x)+25*x^2+200*x+400)),x,method=_RETURNVERBOSE)
 

Output:

ln(ln(((64*x^2+512*x+1024)*ln(x)^2+(608*x^2+5120*x+10752)*ln(x)+1444*x^2+1 
2768*x+28224)/(x^2*ln(x)^2+8*x*ln(x)^2+10*x^2*ln(x)+16*ln(x)^2+80*x*ln(x)+ 
25*x^2+160*ln(x)+200*x+400)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (23) = 46\).

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log \left (\log \left (\frac {4 \, {\left (16 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{2} + 361 \, x^{2} + 8 \, {\left (19 \, x^{2} + 160 \, x + 336\right )} \log \left (x\right ) + 3192 \, x + 7056\right )}}{{\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{2} + 25 \, x^{2} + 10 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right ) + 200 \, x + 400}\right )\right ) \] Input:

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x 
^3+320*x^2+656*x)*log(x)+95*x^3+800*x^2+1680*x)/log(((64*x^2+512*x+1024)*l 
og(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16 
)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="frica 
s")
 

Output:

log(log(4*(16*(x^2 + 8*x + 16)*log(x)^2 + 361*x^2 + 8*(19*x^2 + 160*x + 33 
6)*log(x) + 3192*x + 7056)/((x^2 + 8*x + 16)*log(x)^2 + 25*x^2 + 10*(x^2 + 
 8*x + 16)*log(x) + 200*x + 400)))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (32) = 64\).

Time = 0.77 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.25 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log {\left (\log {\left (\frac {1444 x^{2} + 12768 x + \left (64 x^{2} + 512 x + 1024\right ) \log {\left (x \right )}^{2} + \left (608 x^{2} + 5120 x + 10752\right ) \log {\left (x \right )} + 28224}{25 x^{2} + 200 x + \left (x^{2} + 8 x + 16\right ) \log {\left (x \right )}^{2} + \left (10 x^{2} + 80 x + 160\right ) \log {\left (x \right )} + 400} \right )} \right )} \] Input:

integrate((-16*x*ln(x)+2*x**2-80*x-32)/((4*x**3+32*x**2+64*x)*ln(x)**2+(39 
*x**3+320*x**2+656*x)*ln(x)+95*x**3+800*x**2+1680*x)/ln(((64*x**2+512*x+10 
24)*ln(x)**2+(608*x**2+5120*x+10752)*ln(x)+1444*x**2+12768*x+28224)/((x**2 
+8*x+16)*ln(x)**2+(10*x**2+80*x+160)*ln(x)+25*x**2+200*x+400)),x)
 

Output:

log(log((1444*x**2 + 12768*x + (64*x**2 + 512*x + 1024)*log(x)**2 + (608*x 
**2 + 5120*x + 10752)*log(x) + 28224)/(25*x**2 + 200*x + (x**2 + 8*x + 16) 
*log(x)**2 + (10*x**2 + 80*x + 160)*log(x) + 400)))
 

Maxima [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log \left (\log \left (2\right ) + \log \left (4 \, {\left (x + 4\right )} \log \left (x\right ) + 19 \, x + 84\right ) - \log \left (x + 4\right ) - \log \left (\log \left (x\right ) + 5\right )\right ) \] Input:

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x 
^3+320*x^2+656*x)*log(x)+95*x^3+800*x^2+1680*x)/log(((64*x^2+512*x+1024)*l 
og(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16 
)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="maxim 
a")
 

Output:

log(log(2) + log(4*(x + 4)*log(x) + 19*x + 84) - log(x + 4) - log(log(x) + 
 5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (23) = 46\).

Time = 0.74 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.21 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\log \left (-\log \left (64 \, x^{2} \log \left (x\right )^{2} + 608 \, x^{2} \log \left (x\right ) + 512 \, x \log \left (x\right )^{2} + 1444 \, x^{2} + 5120 \, x \log \left (x\right ) + 1024 \, \log \left (x\right )^{2} + 12768 \, x + 10752 \, \log \left (x\right ) + 28224\right ) + \log \left (x^{2} \log \left (x\right )^{2} + 10 \, x^{2} \log \left (x\right ) + 8 \, x \log \left (x\right )^{2} + 25 \, x^{2} + 80 \, x \log \left (x\right ) + 16 \, \log \left (x\right )^{2} + 200 \, x + 160 \, \log \left (x\right ) + 400\right )\right ) \] Input:

integrate((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x 
^3+320*x^2+656*x)*log(x)+95*x^3+800*x^2+1680*x)/log(((64*x^2+512*x+1024)*l 
og(x)^2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16 
)*log(x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x, algorithm="giac" 
)
 

Output:

log(-log(64*x^2*log(x)^2 + 608*x^2*log(x) + 512*x*log(x)^2 + 1444*x^2 + 51 
20*x*log(x) + 1024*log(x)^2 + 12768*x + 10752*log(x) + 28224) + log(x^2*lo 
g(x)^2 + 10*x^2*log(x) + 8*x*log(x)^2 + 25*x^2 + 80*x*log(x) + 16*log(x)^2 
 + 200*x + 160*log(x) + 400))
 

Mupad [B] (verification not implemented)

Time = 4.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.29 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\ln \left (\ln \left (\frac {12768\,x+{\ln \left (x\right )}^2\,\left (64\,x^2+512\,x+1024\right )+\ln \left (x\right )\,\left (608\,x^2+5120\,x+10752\right )+1444\,x^2+28224}{200\,x+\ln \left (x\right )\,\left (10\,x^2+80\,x+160\right )+{\ln \left (x\right )}^2\,\left (x^2+8\,x+16\right )+25\,x^2+400}\right )\right ) \] Input:

int(-(80*x + 16*x*log(x) - 2*x^2 + 32)/(log((12768*x + log(x)^2*(512*x + 6 
4*x^2 + 1024) + log(x)*(5120*x + 608*x^2 + 10752) + 1444*x^2 + 28224)/(200 
*x + log(x)*(80*x + 10*x^2 + 160) + log(x)^2*(8*x + x^2 + 16) + 25*x^2 + 4 
00))*(1680*x + log(x)^2*(64*x + 32*x^2 + 4*x^3) + 800*x^2 + 95*x^3 + log(x 
)*(656*x + 320*x^2 + 39*x^3))),x)
 

Output:

log(log((12768*x + log(x)^2*(512*x + 64*x^2 + 1024) + log(x)*(5120*x + 608 
*x^2 + 10752) + 1444*x^2 + 28224)/(200*x + log(x)*(80*x + 10*x^2 + 160) + 
log(x)^2*(8*x + x^2 + 16) + 25*x^2 + 400)))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {-32-80 x+2 x^2-16 x \log (x)}{\left (1680 x+800 x^2+95 x^3+\left (656 x+320 x^2+39 x^3\right ) \log (x)+\left (64 x+32 x^2+4 x^3\right ) \log ^2(x)\right ) \log \left (\frac {28224+12768 x+1444 x^2+\left (10752+5120 x+608 x^2\right ) \log (x)+\left (1024+512 x+64 x^2\right ) \log ^2(x)}{400+200 x+25 x^2+\left (160+80 x+10 x^2\right ) \log (x)+\left (16+8 x+x^2\right ) \log ^2(x)}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {64 \mathrm {log}\left (x \right )^{2} x^{2}+512 \mathrm {log}\left (x \right )^{2} x +1024 \mathrm {log}\left (x \right )^{2}+608 \,\mathrm {log}\left (x \right ) x^{2}+5120 \,\mathrm {log}\left (x \right ) x +10752 \,\mathrm {log}\left (x \right )+1444 x^{2}+12768 x +28224}{\mathrm {log}\left (x \right )^{2} x^{2}+8 \mathrm {log}\left (x \right )^{2} x +16 \mathrm {log}\left (x \right )^{2}+10 \,\mathrm {log}\left (x \right ) x^{2}+80 \,\mathrm {log}\left (x \right ) x +160 \,\mathrm {log}\left (x \right )+25 x^{2}+200 x +400}\right )\right ) \] Input:

int((-16*x*log(x)+2*x^2-80*x-32)/((4*x^3+32*x^2+64*x)*log(x)^2+(39*x^3+320 
*x^2+656*x)*log(x)+95*x^3+800*x^2+1680*x)/log(((64*x^2+512*x+1024)*log(x)^ 
2+(608*x^2+5120*x+10752)*log(x)+1444*x^2+12768*x+28224)/((x^2+8*x+16)*log( 
x)^2+(10*x^2+80*x+160)*log(x)+25*x^2+200*x+400)),x)
 

Output:

log(log((64*log(x)**2*x**2 + 512*log(x)**2*x + 1024*log(x)**2 + 608*log(x) 
*x**2 + 5120*log(x)*x + 10752*log(x) + 1444*x**2 + 12768*x + 28224)/(log(x 
)**2*x**2 + 8*log(x)**2*x + 16*log(x)**2 + 10*log(x)*x**2 + 80*log(x)*x + 
160*log(x) + 25*x**2 + 200*x + 400)))