Integrand size = 99, antiderivative size = 29 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=5 \log (x) \left (x (x-\log (-3+x))-\left (-1+x+\frac {1}{16} \log (\log (3))\right )^2\right ) \] Output:
5*ln(x)*(x*(x-ln(-3+x))-(1/16*ln(ln(3))+x-1)^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.07 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=-15 \log (3) \log (-3+x)-5 \log (x)+10 x \log (x)+15 \log \left (1-\frac {x}{3}\right ) \log (x)-5 x \log (-3+x) \log (x)+\frac {5}{8} \log (x) \log (\log (3))-\frac {5}{8} x \log (x) \log (\log (3))-\frac {5}{256} \log (x) \log ^2(\log (3))+15 \operatorname {PolyLog}\left (2,1-\frac {x}{3}\right )+15 \operatorname {PolyLog}\left (2,\frac {x}{3}\right ) \] Input:
Integrate[(3840 - 8960*x + 2560*x^2 + (3840*x - 1280*x^2)*Log[-3 + x] + (- 7680*x + 1280*x^2 + (3840*x - 1280*x^2)*Log[-3 + x])*Log[x] + (-480 + 640* x - 160*x^2 + (480*x - 160*x^2)*Log[x])*Log[Log[3]] + (15 - 5*x)*Log[Log[3 ]]^2)/(-768*x + 256*x^2),x]
Output:
-15*Log[3]*Log[-3 + x] - 5*Log[x] + 10*x*Log[x] + 15*Log[1 - x/3]*Log[x] - 5*x*Log[-3 + x]*Log[x] + (5*Log[x]*Log[Log[3]])/8 - (5*x*Log[x]*Log[Log[3 ]])/8 - (5*Log[x]*Log[Log[3]]^2)/256 + 15*PolyLog[2, 1 - x/3] + 15*PolyLog [2, x/3]
Leaf count is larger than twice the leaf count of optimal. \(110\) vs. \(2(29)=58\).
Time = 0.66 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.79, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2026, 7239, 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.
| \(\displaystyle \int \frac {2560 x^2+\left (3840 x-1280 x^2\right ) \log (x-3)+\left (1280 x^2+\left (3840 x-1280 x^2\right ) \log (x-3)-7680 x\right ) \log (x)+\log (\log (3)) \left (-160 x^2+\left (480 x-160 x^2\right ) \log (x)+640 x-480\right )-8960 x+(15-5 x) \log ^2(\log (3))+3840}{256 x^2-768 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2560 x^2+\left (3840 x-1280 x^2\right ) \log (x-3)+\left (1280 x^2+\left (3840 x-1280 x^2\right ) \log (x-3)-7680 x\right ) \log (x)+\log (\log (3)) \left (-160 x^2+\left (480 x-160 x^2\right ) \log (x)+640 x-480\right )-8960 x+(15-5 x) \log ^2(\log (3))+3840}{x (256 x-768)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (-5 \log (x-3) (\log (x)+1)-\frac {5 \log (x) (x (\log (\log (3))-8)+48-3 \log (\log (3)))}{8 (x-3)}-\frac {5 (\log (\log (3))-16) (32 x-16+\log (\log (3)))}{256 x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -10 x+5 x (\log (x)+1)+\frac {5}{8} x (16-\log (\log (3)))+\frac {5}{8} x (8-\log (\log (3))) \log (x)-\frac {5}{8} x (8-\log (\log (3)))-5 (3-x) \log (x-3)-15 \log (3) \log (x-3)-15 \log (x-3) \log \left (\frac {x}{3}\right )+5 (3-x) \log (x-3) (\log (x)+1)-\frac {5}{256} (16-\log (\log (3)))^2 \log (x)\) |
Input:
Int[(3840 - 8960*x + 2560*x^2 + (3840*x - 1280*x^2)*Log[-3 + x] + (-7680*x + 1280*x^2 + (3840*x - 1280*x^2)*Log[-3 + x])*Log[x] + (-480 + 640*x - 16 0*x^2 + (480*x - 160*x^2)*Log[x])*Log[Log[3]] + (15 - 5*x)*Log[Log[3]]^2)/ (-768*x + 256*x^2),x]
Output:
-10*x - 5*(3 - x)*Log[-3 + x] - 15*Log[3]*Log[-3 + x] - 15*Log[-3 + x]*Log [x/3] + 5*x*(1 + Log[x]) + 5*(3 - x)*Log[-3 + x]*(1 + Log[x]) - (5*x*(8 - Log[Log[3]]))/8 + (5*x*Log[x]*(8 - Log[Log[3]]))/8 + (5*x*(16 - Log[Log[3] ]))/8 - (5*Log[x]*(16 - Log[Log[3]])^2)/256
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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.64 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34
| method | result | size |
| norman | \(\left (-5+\frac {5 \ln \left (\ln \left (3\right )\right )}{8}-\frac {5 \ln \left (\ln \left (3\right )\right )^{2}}{256}\right ) \ln \left (x \right )+\left (10-\frac {5 \ln \left (\ln \left (3\right )\right )}{8}\right ) x \ln \left (x \right )-5 \ln \left (x \right ) \ln \left (-3+x \right ) x\) | \(39\) |
| risch | \(-5 \ln \left (x \right ) \ln \left (-3+x \right ) x -\frac {5 \ln \left (\ln \left (3\right )\right ) \ln \left (x \right ) x}{8}+10 x \ln \left (x \right )-\frac {5 \ln \left (\ln \left (3\right )\right )^{2} \ln \left (x \right )}{256}+\frac {5 \ln \left (\ln \left (3\right )\right ) \ln \left (x \right )}{8}-5 \ln \left (x \right )\) | \(44\) |
| parallelrisch | \(-5 \ln \left (x \right ) \ln \left (-3+x \right ) x -\frac {5 \ln \left (\ln \left (3\right )\right ) \ln \left (x \right ) x}{8}+10 x \ln \left (x \right )-\frac {5 \ln \left (\ln \left (3\right )\right )^{2} \ln \left (x \right )}{256}+\frac {5 \ln \left (\ln \left (3\right )\right ) \ln \left (x \right )}{8}-5 \ln \left (x \right )\) | \(44\) |
| parts | \(-15 \ln \left (-3+x \right )-10 x +\frac {5 \ln \left (\ln \left (3\right )\right ) x}{8}+\frac {5 \left (16-\ln \left (\ln \left (3\right )\right )\right ) \ln \left (x \right ) x}{8}+5 \ln \left (-3+x \right ) x -5 \ln \left (x \right ) \ln \left (-3+x \right ) x -\frac {5 \left (-16+\ln \left (\ln \left (3\right )\right )\right ) \left (32 x +\left (-16+\ln \left (\ln \left (3\right )\right )\right ) \ln \left (x \right )\right )}{256}-5 \left (-3+x \right ) \ln \left (-3+x \right )-15\) | \(74\) |
| default | \(-15 \ln \left (-3+x \right )+\frac {5 \ln \left (\ln \left (3\right )\right ) x}{8}+\frac {5 \left (16-\ln \left (\ln \left (3\right )\right )\right ) \ln \left (x \right ) x}{8}+5 \ln \left (-3+x \right ) x -5 \ln \left (x \right ) \ln \left (-3+x \right ) x -\frac {5 \ln \left (\ln \left (3\right )\right )^{2} \ln \left (x \right )}{256}-5 \ln \left (x \right )-\frac {5 \ln \left (\ln \left (3\right )\right ) \left (x -\ln \left (x \right )\right )}{8}-5 \left (-3+x \right ) \ln \left (-3+x \right )-15\) | \(76\) |
Input:
int(((15-5*x)*ln(ln(3))^2+((-160*x^2+480*x)*ln(x)-160*x^2+640*x-480)*ln(ln (3))+((-1280*x^2+3840*x)*ln(-3+x)+1280*x^2-7680*x)*ln(x)+(-1280*x^2+3840*x )*ln(-3+x)+2560*x^2-8960*x+3840)/(256*x^2-768*x),x,method=_RETURNVERBOSE)
Output:
(-5+5/8*ln(ln(3))-5/256*ln(ln(3))^2)*ln(x)+(10-5/8*ln(ln(3)))*x*ln(x)-5*ln (x)*ln(-3+x)*x
Time = 0.07 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=-\frac {5}{8} \, {\left (x - 1\right )} \log \left (x\right ) \log \left (\log \left (3\right )\right ) - \frac {5}{256} \, \log \left (x\right ) \log \left (\log \left (3\right )\right )^{2} - 5 \, {\left (x \log \left (x - 3\right ) - 2 \, x + 1\right )} \log \left (x\right ) \] Input:
integrate(((15-5*x)*log(log(3))^2+((-160*x^2+480*x)*log(x)-160*x^2+640*x-4 80)*log(log(3))+((-1280*x^2+3840*x)*log(-3+x)+1280*x^2-7680*x)*log(x)+(-12 80*x^2+3840*x)*log(-3+x)+2560*x^2-8960*x+3840)/(256*x^2-768*x),x, algorith m="fricas")
Output:
-5/8*(x - 1)*log(x)*log(log(3)) - 5/256*log(x)*log(log(3))^2 - 5*(x*log(x - 3) - 2*x + 1)*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (24) = 48\).
Time = 1.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.34 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=\left (- 5 x \log {\left (x \right )} - \frac {15}{4}\right ) \log {\left (x - 3 \right )} + \left (- \frac {5 x \log {\left (\log {\left (3 \right )} \right )}}{8} + 10 x\right ) \log {\left (x \right )} - \left (- \frac {5 \log {\left (\log {\left (3 \right )} \right )}}{8} + \frac {5 \log {\left (\log {\left (3 \right )} \right )}^{2}}{256} + 5\right ) \log {\left (x \right )} + \frac {15 \log {\left (x + \frac {-6720 - 15 \log {\left (\log {\left (3 \right )} \right )}^{2} + 480 \log {\left (\log {\left (3 \right )} \right )}}{- 160 \log {\left (\log {\left (3 \right )} \right )} + 5 \log {\left (\log {\left (3 \right )} \right )}^{2} + 2240} \right )}}{4} \] Input:
integrate(((15-5*x)*ln(ln(3))**2+((-160*x**2+480*x)*ln(x)-160*x**2+640*x-4 80)*ln(ln(3))+((-1280*x**2+3840*x)*ln(-3+x)+1280*x**2-7680*x)*ln(x)+(-1280 *x**2+3840*x)*ln(-3+x)+2560*x**2-8960*x+3840)/(256*x**2-768*x),x)
Output:
(-5*x*log(x) - 15/4)*log(x - 3) + (-5*x*log(log(3))/8 + 10*x)*log(x) - (-5 *log(log(3))/8 + 5*log(log(3))**2/256 + 5)*log(x) + 15*log(x + (-6720 - 15 *log(log(3))**2 + 480*log(log(3)))/(-160*log(log(3)) + 5*log(log(3))**2 + 2240))/4
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.55 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=-\frac {5}{8} \, x {\left (\log \left (\log \left (3\right )\right ) - 16\right )} \log \left (x\right ) + \frac {5}{256} \, {\left (\log \left (x - 3\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right )\right )^{2} - \frac {5}{256} \, \log \left (x - 3\right ) \log \left (\log \left (3\right )\right )^{2} + \frac {5}{8} \, x {\left (\log \left (\log \left (3\right )\right ) - 24\right )} - 5 \, {\left (x \log \left (x\right ) - x + 3\right )} \log \left (x - 3\right ) - 5 \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \left (x - 3\right ) + 15 \, \log \left (x - 3\right )^{2} - \frac {5}{8} \, {\left (x + 3 \, \log \left (x - 3\right )\right )} \log \left (\log \left (3\right )\right ) - \frac {5}{8} \, {\left (\log \left (x - 3\right ) - \log \left (x\right )\right )} \log \left (\log \left (3\right )\right ) + \frac {5}{2} \, \log \left (x - 3\right ) \log \left (\log \left (3\right )\right ) + 15 \, x + 15 \, \log \left (x - 3\right ) - 5 \, \log \left (x\right ) \] Input:
integrate(((15-5*x)*log(log(3))^2+((-160*x^2+480*x)*log(x)-160*x^2+640*x-4 80)*log(log(3))+((-1280*x^2+3840*x)*log(-3+x)+1280*x^2-7680*x)*log(x)+(-12 80*x^2+3840*x)*log(-3+x)+2560*x^2-8960*x+3840)/(256*x^2-768*x),x, algorith m="maxima")
Output:
-5/8*x*(log(log(3)) - 16)*log(x) + 5/256*(log(x - 3) - log(x))*log(log(3)) ^2 - 5/256*log(x - 3)*log(log(3))^2 + 5/8*x*(log(log(3)) - 24) - 5*(x*log( x) - x + 3)*log(x - 3) - 5*(x + 3*log(x - 3))*log(x - 3) + 15*log(x - 3)^2 - 5/8*(x + 3*log(x - 3))*log(log(3)) - 5/8*(log(x - 3) - log(x))*log(log( 3)) + 5/2*log(x - 3)*log(log(3)) + 15*x + 15*log(x - 3) - 5*log(x)
Time = 0.12 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=-\frac {5}{8} \, x {\left (\log \left (\log \left (3\right )\right ) - 16\right )} \log \left (x\right ) - 5 \, x \log \left (x - 3\right ) \log \left (x\right ) - \frac {5}{256} \, {\left (\log \left (\log \left (3\right )\right )^{2} - 32 \, \log \left (\log \left (3\right )\right ) + 256\right )} \log \left (x\right ) \] Input:
integrate(((15-5*x)*log(log(3))^2+((-160*x^2+480*x)*log(x)-160*x^2+640*x-4 80)*log(log(3))+((-1280*x^2+3840*x)*log(-3+x)+1280*x^2-7680*x)*log(x)+(-12 80*x^2+3840*x)*log(-3+x)+2560*x^2-8960*x+3840)/(256*x^2-768*x),x, algorith m="giac")
Output:
-5/8*x*(log(log(3)) - 16)*log(x) - 5*x*log(x - 3)*log(x) - 5/256*(log(log( 3))^2 - 32*log(log(3)) + 256)*log(x)
Time = 2.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=-\frac {5\,\ln \left (x\right )\,\left (256\,x\,\ln \left (x-3\right )-32\,\ln \left (\ln \left (3\right )\right )-512\,x+{\ln \left (\ln \left (3\right )\right )}^2+32\,x\,\ln \left (\ln \left (3\right )\right )+256\right )}{256} \] Input:
int(-(log(x - 3)*(3840*x - 1280*x^2) - 8960*x + log(log(3))*(640*x + log(x )*(480*x - 160*x^2) - 160*x^2 - 480) + log(x)*(log(x - 3)*(3840*x - 1280*x ^2) - 7680*x + 1280*x^2) - log(log(3))^2*(5*x - 15) + 2560*x^2 + 3840)/(76 8*x - 256*x^2),x)
Output:
-(5*log(x)*(256*x*log(x - 3) - 32*log(log(3)) - 512*x + log(log(3))^2 + 32 *x*log(log(3)) + 256))/256
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {3840-8960 x+2560 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)+\left (-7680 x+1280 x^2+\left (3840 x-1280 x^2\right ) \log (-3+x)\right ) \log (x)+\left (-480+640 x-160 x^2+\left (480 x-160 x^2\right ) \log (x)\right ) \log (\log (3))+(15-5 x) \log ^2(\log (3))}{-768 x+256 x^2} \, dx=\frac {5 \,\mathrm {log}\left (x \right ) \left (-\mathrm {log}\left (\mathrm {log}\left (3\right )\right )^{2}-32 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right ) x +32 \,\mathrm {log}\left (\mathrm {log}\left (3\right )\right )-256 \,\mathrm {log}\left (x -3\right ) x +512 x -256\right )}{256} \] Input:
int(((15-5*x)*log(log(3))^2+((-160*x^2+480*x)*log(x)-160*x^2+640*x-480)*lo g(log(3))+((-1280*x^2+3840*x)*log(-3+x)+1280*x^2-7680*x)*log(x)+(-1280*x^2 +3840*x)*log(-3+x)+2560*x^2-8960*x+3840)/(256*x^2-768*x),x)
Output:
(5*log(x)*( - log(log(3))**2 - 32*log(log(3))*x + 32*log(log(3)) - 256*log (x - 3)*x + 512*x - 256))/256