Integrand size = 120, antiderivative size = 33 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=\frac {2}{-1+\frac {2}{5} \left (-x^2+\frac {5}{x-x^2}-\frac {1}{\log (4)}\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(33)=66\).
Time = 0.76 (sec) , antiderivative size = 427, normalized size of antiderivative = 12.94 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=\frac {20 \log ^2(4) \left (5 \left (35700 \log ^5(4)+8 \log (16) (8+27 \log (16))-40 \log ^4(4) (99+2155 \log (16))+\log ^2(4) \left (1760+2800 \log (16)-7150 \log ^2(16)\right )-4 \log (4) \left (32+328 \log (16)+795 \log ^2(16)\right )+5 \log ^3(4) \left (1424+3256 \log (16)+6835 \log ^2(16)\right )\right )+2 x^2 \left (39900 \log ^5(4)-175 \log ^4(4) (-92+587 \log (16))+16 \left (-4-13 \log (16)+10 \log ^2(16)\right )+25 \log ^3(4) \left (144-1588 \log (16)+2095 \log ^2(16)\right )-4 \log (4) \left (-192-812 \log (16)+425 \log ^2(16)+600 \log ^3(16)\right )+20 \log ^2(4) \left (256+542 \log (16)-495 \log ^2(16)+1300 \log ^3(16)\right )\right )-4 x \left (20475 \log ^5(4)-50 \log ^4(4) (-81+1495 \log (16))+16 \left (-2-6 \log (16)+15 \log ^2(16)\right )-8 \log (4) \left (-46-145 \log (16)+275 \log ^2(16)\right )+50 \log ^3(4) \left (36+17 \log (16)+955 \log ^2(16)\right )+4 \log ^2(4) \left (712+2030 \log (16)-4175 \log ^2(16)+2000 \log ^3(16)\right )\right )\right )}{\left (10 \log (4)-2 x^3 \log (4)-x (2+5 \log (4))+x^2 (2+5 \log (4))+x^4 \log (16)\right ) \left (28700 \log ^6(4)+16 \log (16) (8+27 \log (16))-4000 \log ^5(4) (1+48 \log (16))-96 \log (4) \log (16) (16+75 \log (16))-8 \log ^2(4) \left (8+1320 \log (16)+2375 \log ^2(16)\right )+5 \log ^4(4) \left (672-8280 \log (16)+69775 \log ^2(16)\right )-8 \log ^3(4) \left (-112+80 \log (16)-17975 \log ^2(16)+32000 \log ^3(16)\right )\right )} \]
Integrate[((100 - 200*x + 40*x^3 - 80*x^4 + 40*x^5)*Log[4]^2)/(4*x^2 - 8*x ^3 + 4*x^4 + (-40*x + 60*x^2 - 40*x^3 + 28*x^4 - 16*x^5 + 8*x^6)*Log[4] + (100 - 100*x + 125*x^2 - 90*x^3 + 85*x^4 - 40*x^5 + 24*x^6 - 8*x^7 + 4*x^8 )*Log[4]^2),x]
(20*Log[4]^2*(5*(35700*Log[4]^5 + 8*Log[16]*(8 + 27*Log[16]) - 40*Log[4]^4 *(99 + 2155*Log[16]) + Log[4]^2*(1760 + 2800*Log[16] - 7150*Log[16]^2) - 4 *Log[4]*(32 + 328*Log[16] + 795*Log[16]^2) + 5*Log[4]^3*(1424 + 3256*Log[1 6] + 6835*Log[16]^2)) + 2*x^2*(39900*Log[4]^5 - 175*Log[4]^4*(-92 + 587*Lo g[16]) + 16*(-4 - 13*Log[16] + 10*Log[16]^2) + 25*Log[4]^3*(144 - 1588*Log [16] + 2095*Log[16]^2) - 4*Log[4]*(-192 - 812*Log[16] + 425*Log[16]^2 + 60 0*Log[16]^3) + 20*Log[4]^2*(256 + 542*Log[16] - 495*Log[16]^2 + 1300*Log[1 6]^3)) - 4*x*(20475*Log[4]^5 - 50*Log[4]^4*(-81 + 1495*Log[16]) + 16*(-2 - 6*Log[16] + 15*Log[16]^2) - 8*Log[4]*(-46 - 145*Log[16] + 275*Log[16]^2) + 50*Log[4]^3*(36 + 17*Log[16] + 955*Log[16]^2) + 4*Log[4]^2*(712 + 2030*L og[16] - 4175*Log[16]^2 + 2000*Log[16]^3))))/((10*Log[4] - 2*x^3*Log[4] - x*(2 + 5*Log[4]) + x^2*(2 + 5*Log[4]) + x^4*Log[16])*(28700*Log[4]^6 + 16* Log[16]*(8 + 27*Log[16]) - 4000*Log[4]^5*(1 + 48*Log[16]) - 96*Log[4]*Log[ 16]*(16 + 75*Log[16]) - 8*Log[4]^2*(8 + 1320*Log[16] + 2375*Log[16]^2) + 5 *Log[4]^4*(672 - 8280*Log[16] + 69775*Log[16]^2) - 8*Log[4]^3*(-112 + 80*L og[16] - 17975*Log[16]^2 + 32000*Log[16]^3)))
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.
| \(\displaystyle \int \frac {\left (40 x^5-80 x^4+40 x^3-200 x+100\right ) \log ^2(4)}{4 x^4-8 x^3+4 x^2+\left (8 x^6-16 x^5+28 x^4-40 x^3+60 x^2-40 x\right ) \log (4)+\left (4 x^8-8 x^7+24 x^6-40 x^5+85 x^4-90 x^3+125 x^2-100 x+100\right ) \log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \log ^2(4) \int \frac {20 \left (2 x^5-4 x^4+2 x^3-10 x+5\right )}{4 x^4-8 x^3+4 x^2+\left (4 x^8-8 x^7+24 x^6-40 x^5+85 x^4-90 x^3+125 x^2-100 x+100\right ) \log ^2(4)-4 \left (-2 x^6+4 x^5-7 x^4+10 x^3-15 x^2+10 x\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 20 \log ^2(4) \int \frac {2 x^5-4 x^4+2 x^3-10 x+5}{4 x^4-8 x^3+4 x^2+\left (4 x^8-8 x^7+24 x^6-40 x^5+85 x^4-90 x^3+125 x^2-100 x+100\right ) \log ^2(4)-4 \left (-2 x^6+4 x^5-7 x^4+10 x^3-15 x^2+10 x\right ) \log (4)}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle 20 \log ^2(4) \int \left (\frac {x-1}{\log (4) \left (\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)\right )}+\frac {-\left ((2+5 \log (4)) x^3\right )+2 (2+5 \log (4)) x^2-(2+25 \log (4)) x+15 \log (4)}{\log (4) \left (\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 20 \log ^2(4) \left (\frac {\left (4+25 \log ^2(4)+\log (4) (20-50 \log (16))-4 \log (16)\right ) \int \frac {x}{\left (\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)\right )^2}dx}{2 \log (4) \log (16)}+\frac {\int \frac {1}{-\log (16) x^4+\log (16) x^3-(2+5 \log (4)) x^2+(2+5 \log (4)) x-10 \log (4)}dx}{\log (4)}-\frac {(4+5 \log (4) (4+5 \log (4)-4 \log (4096))) \int \frac {1}{\left (\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)\right )^2}dx}{4 \log (4) \log (16)}+\frac {(2+5 \log (4)) \log (1024) \int \frac {x^2}{\left (\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)\right )^2}dx}{2 \log (4) \log (16)}+\frac {\int \frac {x}{\log (16) x^4-2 \log (4) x^3+(2+5 \log (4)) x^2-(2+5 \log (4)) x+10 \log (4)}dx}{\log (4)}+\frac {2+5 \log (4)}{4 \log (4) \log (16) \left (x^4 \log (16)-2 x^3 \log (4)+x^2 (2+5 \log (4))-x (2+5 \log (4))+10 \log (4)\right )}\right )\) |
Int[((100 - 200*x + 40*x^3 - 80*x^4 + 40*x^5)*Log[4]^2)/(4*x^2 - 8*x^3 + 4 *x^4 + (-40*x + 60*x^2 - 40*x^3 + 28*x^4 - 16*x^5 + 8*x^6)*Log[4] + (100 - 100*x + 125*x^2 - 90*x^3 + 85*x^4 - 40*x^5 + 24*x^6 - 8*x^7 + 4*x^8)*Log[ 4]^2),x]
3.4.97.3.1 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_.)*(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]
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45
| method | result | size |
| gosper | \(-\frac {10 x \left (-1+x \right ) \ln \left (2\right )}{2 x^{4} \ln \left (2\right )-2 x^{3} \ln \left (2\right )+5 x^{2} \ln \left (2\right )-5 x \ln \left (2\right )+x^{2}+10 \ln \left (2\right )-x}\) | \(48\) |
| parallelrisch | \(\frac {2 \ln \left (2\right ) \left (-5 x^{2}+5 x \right )}{2 x^{4} \ln \left (2\right )-2 x^{3} \ln \left (2\right )+5 x^{2} \ln \left (2\right )-5 x \ln \left (2\right )+x^{2}+10 \ln \left (2\right )-x}\) | \(53\) |
| norman | \(\frac {10 x \ln \left (2\right )-10 x^{2} \ln \left (2\right )}{2 x^{4} \ln \left (2\right )-2 x^{3} \ln \left (2\right )+5 x^{2} \ln \left (2\right )-5 x \ln \left (2\right )+x^{2}+10 \ln \left (2\right )-x}\) | \(54\) |
| default | \(\frac {20 \ln \left (2\right )^{2} \left (-\frac {x^{2}}{4 \ln \left (2\right )}+\frac {x}{4 \ln \left (2\right )}\right )}{x^{4} \ln \left (2\right )-x^{3} \ln \left (2\right )+\frac {5 x^{2} \ln \left (2\right )}{2}-\frac {5 x \ln \left (2\right )}{2}+\frac {x^{2}}{2}+5 \ln \left (2\right )-\frac {x}{2}}\) | \(64\) |
| risch | \(\frac {4 \ln \left (2\right )^{2} \left (-\frac {5 x^{2}}{4 \ln \left (2\right )}+\frac {5 x}{4 \ln \left (2\right )}\right )}{x^{4} \ln \left (2\right )-x^{3} \ln \left (2\right )+\frac {5 x^{2} \ln \left (2\right )}{2}-\frac {5 x \ln \left (2\right )}{2}+\frac {x^{2}}{2}+5 \ln \left (2\right )-\frac {x}{2}}\) | \(64\) |
int(4*(40*x^5-80*x^4+40*x^3-200*x+100)*ln(2)^2/(4*(4*x^8-8*x^7+24*x^6-40*x ^5+85*x^4-90*x^3+125*x^2-100*x+100)*ln(2)^2+2*(8*x^6-16*x^5+28*x^4-40*x^3+ 60*x^2-40*x)*ln(2)+4*x^4-8*x^3+4*x^2),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=-\frac {10 \, {\left (x^{2} - x\right )} \log \left (2\right )}{x^{2} + {\left (2 \, x^{4} - 2 \, x^{3} + 5 \, x^{2} - 5 \, x + 10\right )} \log \left (2\right ) - x} \]
integrate(4*(40*x^5-80*x^4+40*x^3-200*x+100)*log(2)^2/(4*(4*x^8-8*x^7+24*x ^6-40*x^5+85*x^4-90*x^3+125*x^2-100*x+100)*log(2)^2+2*(8*x^6-16*x^5+28*x^4 -40*x^3+60*x^2-40*x)*log(2)+4*x^4-8*x^3+4*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (22) = 44\).
Time = 1.96 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=\frac {- 10 x^{2} \log {\left (2 \right )} + 10 x \log {\left (2 \right )}}{2 x^{4} \log {\left (2 \right )} - 2 x^{3} \log {\left (2 \right )} + x^{2} \cdot \left (1 + 5 \log {\left (2 \right )}\right ) + x \left (- 5 \log {\left (2 \right )} - 1\right ) + 10 \log {\left (2 \right )}} \]
integrate(4*(40*x**5-80*x**4+40*x**3-200*x+100)*ln(2)**2/(4*(4*x**8-8*x**7 +24*x**6-40*x**5+85*x**4-90*x**3+125*x**2-100*x+100)*ln(2)**2+2*(8*x**6-16 *x**5+28*x**4-40*x**3+60*x**2-40*x)*ln(2)+4*x**4-8*x**3+4*x**2),x)
(-10*x**2*log(2) + 10*x*log(2))/(2*x**4*log(2) - 2*x**3*log(2) + x**2*(1 + 5*log(2)) + x*(-5*log(2) - 1) + 10*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=-\frac {10 \, {\left (x^{2} - x\right )} \log \left (2\right )^{2}}{2 \, x^{4} \log \left (2\right )^{2} - 2 \, x^{3} \log \left (2\right )^{2} + {\left (5 \, \log \left (2\right )^{2} + \log \left (2\right )\right )} x^{2} - {\left (5 \, \log \left (2\right )^{2} + \log \left (2\right )\right )} x + 10 \, \log \left (2\right )^{2}} \]
integrate(4*(40*x^5-80*x^4+40*x^3-200*x+100)*log(2)^2/(4*(4*x^8-8*x^7+24*x ^6-40*x^5+85*x^4-90*x^3+125*x^2-100*x+100)*log(2)^2+2*(8*x^6-16*x^5+28*x^4 -40*x^3+60*x^2-40*x)*log(2)+4*x^4-8*x^3+4*x^2),x, algorithm=\
-10*(x^2 - x)*log(2)^2/(2*x^4*log(2)^2 - 2*x^3*log(2)^2 + (5*log(2)^2 + lo g(2))*x^2 - (5*log(2)^2 + log(2))*x + 10*log(2)^2)
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=-\frac {10 \, {\left (x^{2} - x\right )} \log \left (2\right )}{2 \, x^{4} \log \left (2\right ) - 2 \, x^{3} \log \left (2\right ) + 5 \, x^{2} \log \left (2\right ) + x^{2} - 5 \, x \log \left (2\right ) - x + 10 \, \log \left (2\right )} \]
integrate(4*(40*x^5-80*x^4+40*x^3-200*x+100)*log(2)^2/(4*(4*x^8-8*x^7+24*x ^6-40*x^5+85*x^4-90*x^3+125*x^2-100*x+100)*log(2)^2+2*(8*x^6-16*x^5+28*x^4 -40*x^3+60*x^2-40*x)*log(2)+4*x^4-8*x^3+4*x^2),x, algorithm=\
-10*(x^2 - x)*log(2)/(2*x^4*log(2) - 2*x^3*log(2) + 5*x^2*log(2) + x^2 - 5 *x*log(2) - x + 10*log(2))
Time = 41.85 (sec) , antiderivative size = 256790, normalized size of antiderivative = 7781.52 \[ \int \frac {\left (100-200 x+40 x^3-80 x^4+40 x^5\right ) \log ^2(4)}{4 x^2-8 x^3+4 x^4+\left (-40 x+60 x^2-40 x^3+28 x^4-16 x^5+8 x^6\right ) \log (4)+\left (100-100 x+125 x^2-90 x^3+85 x^4-40 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2(4)} \, dx=\text {Too large to display} \]
int((4*log(2)^2*(40*x^3 - 200*x - 80*x^4 + 40*x^5 + 100))/(4*x^2 - 2*log(2 )*(40*x - 60*x^2 + 40*x^3 - 28*x^4 + 16*x^5 - 8*x^6) - 8*x^3 + 4*x^4 + 4*l og(2)^2*(125*x^2 - 100*x - 90*x^3 + 85*x^4 - 40*x^5 + 24*x^6 - 8*x^7 + 4*x ^8 + 100)),x)
symsum(log(root(42626496416000*z^8*log(2)^9*log(16)*log(256)^4 - 201020737 28000*z^8*log(2)^9*log(16)^2*log(256)^3 - 3219040000*z^8*log(2)^5*log(16)^ 5*log(256) - 14574377200000*z^8*log(2)^10*log(16)^3*log(256)^3 - 464816792 00000*z^8*log(2)^9*log(16)^3*log(256)^3 + 20180869120000*z^8*log(2)^10*log (16)^2*log(256) - 6059008000000*z^8*log(2)^8*log(16)^6*log(256) - 18616650 0000*z^8*log(2)^7*log(16)^2*log(256)^5 + 10014080000*z^8*log(2)^5*log(16)^ 5*log(256)^2 + 1350732800*z^8*log(2)^5*log(16)^4*log(256)^2 + 39960000000* z^8*log(2)^6*log(16)^6*log(256)^2 - 6376625600000*z^8*log(2)^8*log(16)^5*l og(256) + 3250114688000000*z^8*log(2)^14*log(16)^2*log(256)^2 - 1129214400 00*z^8*log(2)^6*log(16)^4*log(256)^3 + 266844855500800*z^8*log(2)^10*log(1 6)*log(256)^2 - 148827148000*z^8*log(2)^7*log(16)*log(256)^5 - 35930195968 000*z^8*log(2)^9*log(16)*log(256)^3 - 18369800000*z^8*log(2)^6*log(16)^2*l og(256)^5 - 434954240000*z^8*log(2)^9*log(16)^2*log(256)^2 - 5221984501760 000*z^8*log(2)^12*log(16)*log(256) - 53903360*z^8*log(2)^6*log(16)*log(256 ) - 510818755200000*z^8*log(2)^11*log(16)^3*log(256)^2 - 163840*z^8*log(2) ^5*log(16)*log(256) - 10240*z^8*log(2)*log(16)^6*log(256) - 640*z^8*log(2) *log(16)^5*log(256) - 513769920000*z^8*log(2)^7*log(16)^5*log(256) + 54166 500000*z^8*log(2)^7*log(16)*log(256)^6 + 3106973376000000*z^8*log(2)^12*lo g(16)^3*log(256) + 2595148032000*z^8*log(2)^8*log(16)^3*log(256)^2 - 56822 128640*z^8*log(2)^7*log(16)^2*log(256)^2 - 7435574880000*z^8*log(2)^8*l...