\(\int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+(-656+1024 x-1136 x^2+1024 x^3-256 x^4) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+(50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7) \log (\log (4))+(8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6) \log ^2(\log (4))} \, dx\) [255]

Optimal result

Integrand size = 150, antiderivative size = 35 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {4}{5 x \left (4+\frac {25 x^2}{4 \left (x-x^2\right )^2}\right ) (3+x+\log (\log (4)))} \] Output:

4/5/x/(3+x+ln(2*ln(2)))/(4+25/4*x^2/(-x^2+x)^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 (-1+x)^2}{5 x \left (41-32 x+16 x^2\right ) (3+x+\log (\log (4)))} \] Input:

Integrate[(-1968 + 1760*x - 560*x^2 + 1280*x^4 - 512*x^5 + (-656 + 1024*x 
- 1136*x^2 + 1024*x^3 - 256*x^4)*Log[Log[4]])/(75645*x^2 - 67650*x^3 + 348 
05*x^4 + 10880*x^5 - 7520*x^6 + 2560*x^7 + 1280*x^8 + (50430*x^2 - 61910*x 
^3 + 43840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7)*Log[Log[4]] + (8405*x^2 - 
 13120*x^3 + 11680*x^4 - 5120*x^5 + 1280*x^6)*Log[Log[4]]^2),x]
 

Output:

(16*(-1 + x)^2)/(5*x*(41 - 32*x + 16*x^2)*(3 + x + Log[Log[4]]))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(35)=70\).

Time = 0.75 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2026, 2462, 2009}

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.

\(\displaystyle \int \frac {-512 x^5+1280 x^4-560 x^2+\left (-256 x^4+1024 x^3-1136 x^2+1024 x-656\right ) \log (\log (4))+1760 x-1968}{1280 x^8+2560 x^7-7520 x^6+10880 x^5+34805 x^4-67650 x^3+75645 x^2+\left (1280 x^6-5120 x^5+11680 x^4-13120 x^3+8405 x^2\right ) \log ^2(\log (4))+\left (2560 x^7-2560 x^6-7360 x^5+43840 x^4-61910 x^3+50430 x^2\right ) \log (\log (4))} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {-512 x^5+1280 x^4-560 x^2+\left (-256 x^4+1024 x^3-1136 x^2+1024 x-656\right ) \log (\log (4))+1760 x-1968}{x^2 \left (1280 x^6+2560 x^5 (1+\log (\log (4)))-160 x^4 \left (47-8 \log ^2(\log (4))+16 \log (\log (4))\right )+320 x^3 \left (34-16 \log ^2(\log (4))-23 \log (\log (4))\right )+5 x^2 \left (6961+2336 \log ^2(\log (4))+8768 \log (\log (4))\right )-410 x (3+\log (\log (4))) (55+32 \log (\log (4)))+8405 (3+\log (\log (4)))^2\right )}dx\)

\(\Big \downarrow \) 2462

\(\displaystyle \int \left (\frac {2560 (x (39+16 \log (\log (4)))+86+9 \log (\log (4)))}{41 \left (16 x^2-32 x+41\right )^2 \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {1280 (5+\log (\log (4)))}{41 \left (16 x^2-32 x+41\right ) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {16}{205 x^2 (3+\log (\log (4)))}+\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right ) (x+3+\log (\log (4)))^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {80 (-16 x (5+\log (\log (4)))+119+32 \log (\log (4)))}{41 \left (16 x^2-32 x+41\right ) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right )}-\frac {16 (4+\log (\log (4)))^2}{5 (3+\log (\log (4))) \left (281+16 \log ^2(\log (4))+128 \log (\log (4))\right ) (x+3+\log (\log (4)))}+\frac {16}{205 x (3+\log (\log (4)))}\)

Input:

Int[(-1968 + 1760*x - 560*x^2 + 1280*x^4 - 512*x^5 + (-656 + 1024*x - 1136 
*x^2 + 1024*x^3 - 256*x^4)*Log[Log[4]])/(75645*x^2 - 67650*x^3 + 34805*x^4 
 + 10880*x^5 - 7520*x^6 + 2560*x^7 + 1280*x^8 + (50430*x^2 - 61910*x^3 + 4 
3840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7)*Log[Log[4]] + (8405*x^2 - 13120 
*x^3 + 11680*x^4 - 5120*x^5 + 1280*x^6)*Log[Log[4]]^2),x]
 

Output:

16/(205*x*(3 + Log[Log[4]])) - (16*(4 + Log[Log[4]])^2)/(5*(3 + Log[Log[4] 
])*(3 + x + Log[Log[4]])*(281 + 128*Log[Log[4]] + 16*Log[Log[4]]^2)) - (80 
*(119 + 32*Log[Log[4]] - 16*x*(5 + Log[Log[4]])))/(41*(41 - 32*x + 16*x^2) 
*(281 + 128*Log[Log[4]] + 16*Log[Log[4]]^2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

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])
 

rule 2462

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] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06

method	result	size
norman	\(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{x \left (16 x^{2}-32 x +41\right ) \left (3+x +\ln \left (2 \ln \left (2\right )\right )\right )}\)	\(37\)
gosper	\(\frac {16 \left (-1+x \right )^{2}}{5 x \left (16 x^{2} \ln \left (2 \ln \left (2\right )\right )+16 x^{3}-32 x \ln \left (2 \ln \left (2\right )\right )+16 x^{2}+41 \ln \left (2 \ln \left (2\right )\right )-55 x +123\right )}\)	\(53\)
parallelrisch	\(\frac {256 x^{2}-512 x +256}{80 x \left (16 x^{2} \ln \left (2 \ln \left (2\right )\right )+16 x^{3}-32 x \ln \left (2 \ln \left (2\right )\right )+16 x^{2}+41 \ln \left (2 \ln \left (2\right )\right )-55 x +123\right )}\)	\(58\)
risch	\(\frac {\frac {16}{5}-\frac {32}{5} x +\frac {16}{5} x^{2}}{\left (16 x^{2} \ln \left (\ln \left (2\right )\right )+16 x^{2} \ln \left (2\right )+16 x^{3}-32 x \ln \left (\ln \left (2\right )\right )-32 x \ln \left (2\right )+16 x^{2}+41 \ln \left (\ln \left (2\right )\right )+41 \ln \left (2\right )-55 x +123\right ) x}\)	\(68\)
default	\(\frac {16}{5 \left (41 \ln \left (2 \ln \left (2\right )\right )+123\right ) x}-\frac {1280 \left (\left (-\frac {5}{16}-\frac {\ln \left (2 \ln \left (2\right )\right )}{16}\right ) x +\frac {119}{256}+\frac {\ln \left (2 \ln \left (2\right )\right )}{8}\right )}{41 \left (16 \ln \left (2 \ln \left (2\right )\right )^{2}+128 \ln \left (2 \ln \left (2\right )\right )+281\right ) \left (x^{2}-2 x +\frac {41}{16}\right )}-\frac {16 \left (\ln \left (2 \ln \left (2\right )\right )^{2}+8 \ln \left (2 \ln \left (2\right )\right )+16\right )}{5 \left (16 \ln \left (2 \ln \left (2\right )\right )^{2}+128 \ln \left (2 \ln \left (2\right )\right )+281\right ) \left (\ln \left (2 \ln \left (2\right )\right )+3\right ) \left (3+x +\ln \left (2 \ln \left (2\right )\right )\right )}\)	\(127\)

Input:

int(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*ln(2*ln(2))-512*x^5+1280*x^4- 
560*x^2+1760*x-1968)/((1280*x^6-5120*x^5+11680*x^4-13120*x^3+8405*x^2)*ln( 
2*ln(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430*x^2)*ln(2 
*ln(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+75645*x^2 
),x,method=_RETURNVERBOSE)
 

Output:

(16/5-32/5*x+16/5*x^2)/x/(16*x^2-32*x+41)/(3+x+ln(2*ln(2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} - 55 \, x^{2} + {\left (16 \, x^{3} - 32 \, x^{2} + 41 \, x\right )} \log \left (2 \, \log \left (2\right )\right ) + 123 \, x\right )}} \] Input:

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1 
280*x^4-560*x^2+1760*x-1968)/((1280*x^6-5120*x^5+11680*x^4-13120*x^3+8405* 
x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430 
*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x 
^3+75645*x^2),x, algorithm="fricas")
 

Output:

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3 - 55*x^2 + (16*x^3 - 32*x^2 + 41*x)* 
log(2*log(2)) + 123*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (29) = 58\).

Time = 34.58 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.89 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=- \frac {- 16 x^{2} + 32 x - 16}{80 x^{4} + x^{3} \cdot \left (80 \log {\left (\log {\left (2 \right )} \right )} + 80 \log {\left (2 \right )} + 80\right ) + x^{2} \left (-275 - 160 \log {\left (2 \right )} - 160 \log {\left (\log {\left (2 \right )} \right )}\right ) + x \left (205 \log {\left (\log {\left (2 \right )} \right )} + 205 \log {\left (2 \right )} + 615\right )} \] Input:

integrate(((-256*x**4+1024*x**3-1136*x**2+1024*x-656)*ln(2*ln(2))-512*x**5 
+1280*x**4-560*x**2+1760*x-1968)/((1280*x**6-5120*x**5+11680*x**4-13120*x* 
*3+8405*x**2)*ln(2*ln(2))**2+(2560*x**7-2560*x**6-7360*x**5+43840*x**4-619 
10*x**3+50430*x**2)*ln(2*ln(2))+1280*x**8+2560*x**7-7520*x**6+10880*x**5+3 
4805*x**4-67650*x**3+75645*x**2),x)
 

Output:

-(-16*x**2 + 32*x - 16)/(80*x**4 + x**3*(80*log(log(2)) + 80*log(2) + 80) 
+ x**2*(-275 - 160*log(2) - 160*log(log(2))) + x*(205*log(log(2)) + 205*lo 
g(2) + 615))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} {\left (\log \left (2 \, \log \left (2\right )\right ) + 1\right )} - x^{2} {\left (32 \, \log \left (2 \, \log \left (2\right )\right ) + 55\right )} + 41 \, x {\left (\log \left (2 \, \log \left (2\right )\right ) + 3\right )}\right )}} \] Input:

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1 
280*x^4-560*x^2+1760*x-1968)/((1280*x^6-5120*x^5+11680*x^4-13120*x^3+8405* 
x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430 
*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x 
^3+75645*x^2),x, algorithm="maxima")
 

Output:

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3*(log(2*log(2)) + 1) - x^2*(32*log(2* 
log(2)) + 55) + 41*x*(log(2*log(2)) + 3))
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.69 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {16 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, {\left (16 \, x^{4} + 16 \, x^{3} \log \left (2 \, \log \left (2\right )\right ) + 16 \, x^{3} - 32 \, x^{2} \log \left (2 \, \log \left (2\right )\right ) - 55 \, x^{2} + 41 \, x \log \left (2 \, \log \left (2\right )\right ) + 123 \, x\right )}} \] Input:

integrate(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1 
280*x^4-560*x^2+1760*x-1968)/((1280*x^6-5120*x^5+11680*x^4-13120*x^3+8405* 
x^2)*log(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430 
*x^2)*log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x 
^3+75645*x^2),x, algorithm="giac")
 

Output:

16/5*(x^2 - 2*x + 1)/(16*x^4 + 16*x^3*log(2*log(2)) + 16*x^3 - 32*x^2*log( 
2*log(2)) - 55*x^2 + 41*x*log(2*log(2)) + 123*x)
 

Mupad [B] (verification not implemented)

Time = 6.62 (sec) , antiderivative size = 205, normalized size of antiderivative = 5.86 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {\frac {\left (\ln \left ({\ln \left (4\right )}^{1121190}\right )+738961\,{\ln \left (\ln \left (4\right )\right )}^2+261056\,{\ln \left (\ln \left (4\right )\right )}^3+52256\,{\ln \left (\ln \left (4\right )\right )}^4+5632\,{\ln \left (\ln \left (4\right )\right )}^5+256\,{\ln \left (\ln \left (4\right )\right )}^6+710649\right )\,x^2}{5\,{\left (\ln \left (\ln \left (4\right )\right )+3\right )}^2\,\left (\ln \left ({\ln \left (4\right )}^{71936}\right )+25376\,{\ln \left (\ln \left (4\right )\right )}^2+4096\,{\ln \left (\ln \left (4\right )\right )}^3+256\,{\ln \left (\ln \left (4\right )\right )}^4+78961\right )}-\frac {\left (\ln \left ({\ln \left (4\right )}^{2242380}\right )+1477922\,{\ln \left (\ln \left (4\right )\right )}^2+522112\,{\ln \left (\ln \left (4\right )\right )}^3+104512\,{\ln \left (\ln \left (4\right )\right )}^4+11264\,{\ln \left (\ln \left (4\right )\right )}^5+512\,{\ln \left (\ln \left (4\right )\right )}^6+1421298\right )\,x}{5\,{\left (\ln \left (\ln \left (4\right )\right )+3\right )}^2\,\left (\ln \left ({\ln \left (4\right )}^{71936}\right )+25376\,{\ln \left (\ln \left (4\right )\right )}^2+4096\,{\ln \left (\ln \left (4\right )\right )}^3+256\,{\ln \left (\ln \left (4\right )\right )}^4+78961\right )}+\frac {1}{5}}{x^4+\left (\ln \left (\ln \left (4\right )\right )+1\right )\,x^3+\left (-\ln \left ({\ln \left (4\right )}^2\right )-\frac {55}{16}\right )\,x^2+\left (\ln \left ({\ln \left (4\right )}^{41/16}\right )+\frac {123}{16}\right )\,x} \] Input:

int(-(log(2*log(2))*(1136*x^2 - 1024*x - 1024*x^3 + 256*x^4 + 656) - 1760* 
x + 560*x^2 - 1280*x^4 + 512*x^5 + 1968)/(log(2*log(2))*(50430*x^2 - 61910 
*x^3 + 43840*x^4 - 7360*x^5 - 2560*x^6 + 2560*x^7) + log(2*log(2))^2*(8405 
*x^2 - 13120*x^3 + 11680*x^4 - 5120*x^5 + 1280*x^6) + 75645*x^2 - 67650*x^ 
3 + 34805*x^4 + 10880*x^5 - 7520*x^6 + 2560*x^7 + 1280*x^8),x)
 

Output:

((x^2*(log(log(4)^1121190) + 738961*log(log(4))^2 + 261056*log(log(4))^3 + 
 52256*log(log(4))^4 + 5632*log(log(4))^5 + 256*log(log(4))^6 + 710649))/( 
5*(log(log(4)) + 3)^2*(log(log(4)^71936) + 25376*log(log(4))^2 + 4096*log( 
log(4))^3 + 256*log(log(4))^4 + 78961)) - (x*(log(log(4)^2242380) + 147792 
2*log(log(4))^2 + 522112*log(log(4))^3 + 104512*log(log(4))^4 + 11264*log( 
log(4))^5 + 512*log(log(4))^6 + 1421298))/(5*(log(log(4)) + 3)^2*(log(log( 
4)^71936) + 25376*log(log(4))^2 + 4096*log(log(4))^3 + 256*log(log(4))^4 + 
 78961)) + 1/5)/(x^3*(log(log(4)) + 1) + x*(log(log(4)^(41/16)) + 123/16) 
- x^2*(log(log(4)^2) + 55/16) + x^4)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.57 \[ \int \frac {-1968+1760 x-560 x^2+1280 x^4-512 x^5+\left (-656+1024 x-1136 x^2+1024 x^3-256 x^4\right ) \log (\log (4))}{75645 x^2-67650 x^3+34805 x^4+10880 x^5-7520 x^6+2560 x^7+1280 x^8+\left (50430 x^2-61910 x^3+43840 x^4-7360 x^5-2560 x^6+2560 x^7\right ) \log (\log (4))+\left (8405 x^2-13120 x^3+11680 x^4-5120 x^5+1280 x^6\right ) \log ^2(\log (4))} \, dx=\frac {\frac {16}{5} x^{2}-\frac {32}{5} x +\frac {16}{5}}{x \left (16 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x^{2}-32 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x +41 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )+16 x^{3}+16 x^{2}-55 x +123\right )} \] Input:

int(((-256*x^4+1024*x^3-1136*x^2+1024*x-656)*log(2*log(2))-512*x^5+1280*x^ 
4-560*x^2+1760*x-1968)/((1280*x^6-5120*x^5+11680*x^4-13120*x^3+8405*x^2)*l 
og(2*log(2))^2+(2560*x^7-2560*x^6-7360*x^5+43840*x^4-61910*x^3+50430*x^2)* 
log(2*log(2))+1280*x^8+2560*x^7-7520*x^6+10880*x^5+34805*x^4-67650*x^3+756 
45*x^2),x)
 

Output:

(16*(x**2 - 2*x + 1))/(5*x*(16*log(2*log(2))*x**2 - 32*log(2*log(2))*x + 4 
1*log(2*log(2)) + 16*x**3 + 16*x**2 - 55*x + 123))