Integrand size = 103, antiderivative size = 27 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=4 \left (x+\frac {x^2 (1+x)}{2 \left (5+\log (2)-\frac {4 x}{\log (x)}\right )}\right ) \] Output:
2*x^2*(1+x)/(ln(2)+5-4*x/ln(x))+4*x
\[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=\int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx \] Input:
Integrate[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (100 + 20*x + 30*x^2 + (40 + 4*x + 6*x^2)*Log[2] + 4*Log[2]^2)*Log[x]^2 )/(16*x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Log[ x]^2),x]
Output:
Integrate[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (100 + 20*x + 30*x^2 + (40 + 4*x + 6*x^2)*Log[2] + 4*Log[2]^2)*Log[x]^2 )/(16*x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Log[ x]^2), x]
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.
| \(\displaystyle \int \frac {-8 x^3+56 x^2+\left (30 x^2+\left (6 x^2+4 x+40\right ) \log (2)+20 x+100+4 \log ^2(2)\right ) \log ^2(x)+\left (-16 x^3-8 x^2-160 x-32 x \log (2)\right ) \log (x)}{16 x^2+\left (25+\log ^2(2)+10 \log (2)\right ) \log ^2(x)+(-40 x-8 x \log (2)) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-8 x^3+56 x^2+\left (30 x^2+\left (6 x^2+4 x+40\right ) \log (2)+20 x+100+4 \log ^2(2)\right ) \log ^2(x)+\left (-16 x^3-8 x^2-160 x-32 x \log (2)\right ) \log (x)}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 (-4 x-3) x^2}{(5+\log (2)) \left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )}+\frac {8 (x+1) x^2 (4 x-5-\log (2))}{(5+\log (2)) \left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}+\frac {6 x^2+4 x+20+\log (16)}{5+\log (2)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32 \int \frac {x^4}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}dx}{5+\log (2)}-\frac {8 (1+\log (2)) \int \frac {x^3}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}dx}{5+\log (2)}+\frac {32 \int \frac {x^3}{5 \left (1+\frac {\log (2)}{5}\right ) \log (x)-4 x}dx}{5+\log (2)}-8 \int \frac {x^2}{\left (4 x-5 \left (1+\frac {\log (2)}{5}\right ) \log (x)\right )^2}dx+\frac {24 \int \frac {x^2}{5 \left (1+\frac {\log (2)}{5}\right ) \log (x)-4 x}dx}{5+\log (2)}+\frac {2 x^3}{5+\log (2)}+\frac {2 x^2}{5+\log (2)}+\frac {x (20+\log (16))}{5+\log (2)}\) |
Input:
Int[(56*x^2 - 8*x^3 + (-160*x - 8*x^2 - 16*x^3 - 32*x*Log[2])*Log[x] + (10 0 + 20*x + 30*x^2 + (40 + 4*x + 6*x^2)*Log[2] + 4*Log[2]^2)*Log[x]^2)/(16* x^2 + (-40*x - 8*x*Log[2])*Log[x] + (25 + 10*Log[2] + Log[2]^2)*Log[x]^2), x]
Output:
$Aborted
Time = 2.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(\frac {2 x^{2} \ln \left (x \right )+2 x^{3} \ln \left (x \right )+2 \left (10+2 \ln \left (2\right )\right ) \ln \left (x \right ) x -16 x^{2}}{\ln \left (2\right ) \ln \left (x \right )+5 \ln \left (x \right )-4 x}\) | \(46\) |
| norman | \(\frac {\left (4 \ln \left (2\right )+20\right ) x \ln \left (x \right )-16 x^{2}+2 x^{2} \ln \left (x \right )+2 x^{3} \ln \left (x \right )}{\ln \left (2\right ) \ln \left (x \right )+5 \ln \left (x \right )-4 x}\) | \(47\) |
| risch | \(\frac {2 x \left (x^{2}+2 \ln \left (2\right )+x +10\right )}{\ln \left (2\right )+5}+\frac {8 \left (1+x \right ) x^{3}}{\left (\ln \left (2\right )+5\right ) \left (\ln \left (2\right ) \ln \left (x \right )+5 \ln \left (x \right )-4 x \right )}\) | \(50\) |
| parallelrisch | \(\frac {8 x^{3} \ln \left (x \right )+16 x \ln \left (2\right ) \ln \left (x \right )+8 x^{2} \ln \left (x \right )-64 x^{2}+80 x \ln \left (x \right )}{4 \ln \left (2\right ) \ln \left (x \right )+20 \ln \left (x \right )-16 x}\) | \(50\) |
Input:
int(((4*ln(2)^2+(6*x^2+4*x+40)*ln(2)+30*x^2+20*x+100)*ln(x)^2+(-32*x*ln(2) -16*x^3-8*x^2-160*x)*ln(x)-8*x^3+56*x^2)/((ln(2)^2+10*ln(2)+25)*ln(x)^2+(- 8*x*ln(2)-40*x)*ln(x)+16*x^2),x,method=_RETURNVERBOSE)
Output:
2*(x^2*ln(x)+x^3*ln(x)+(10+2*ln(2))*ln(x)*x-8*x^2)/(ln(2)*ln(x)+5*ln(x)-4* x)
Time = 0.07 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=-\frac {2 \, {\left (8 \, x^{2} - {\left (x^{3} + x^{2} + 2 \, x \log \left (2\right ) + 10 \, x\right )} \log \left (x\right )\right )}}{{\left (\log \left (2\right ) + 5\right )} \log \left (x\right ) - 4 \, x} \] Input:
integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-3 2*x*log(2)-16*x^3-8*x^2-160*x)*log(x)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+2 5)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="fricas")
Output:
-2*(8*x^2 - (x^3 + x^2 + 2*x*log(2) + 10*x)*log(x))/((log(2) + 5)*log(x) - 4*x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.15 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=\frac {2 x^{3}}{\log {\left (2 \right )} + 5} + \frac {2 x^{2}}{\log {\left (2 \right )} + 5} + 4 x + \frac {8 x^{4} + 8 x^{3}}{- 20 x - 4 x \log {\left (2 \right )} + \left (\log {\left (2 \right )}^{2} + 10 \log {\left (2 \right )} + 25\right ) \log {\left (x \right )}} \] Input:
integrate(((4*ln(2)**2+(6*x**2+4*x+40)*ln(2)+30*x**2+20*x+100)*ln(x)**2+(- 32*x*ln(2)-16*x**3-8*x**2-160*x)*ln(x)-8*x**3+56*x**2)/((ln(2)**2+10*ln(2) +25)*ln(x)**2+(-8*x*ln(2)-40*x)*ln(x)+16*x**2),x)
Output:
2*x**3/(log(2) + 5) + 2*x**2/(log(2) + 5) + 4*x + (8*x**4 + 8*x**3)/(-20*x - 4*x*log(2) + (log(2)**2 + 10*log(2) + 25)*log(x))
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=-\frac {2 \, {\left (8 \, x^{2} - {\left (x^{3} + x^{2} + 2 \, x {\left (\log \left (2\right ) + 5\right )}\right )} \log \left (x\right )\right )}}{{\left (\log \left (2\right ) + 5\right )} \log \left (x\right ) - 4 \, x} \] Input:
integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-3 2*x*log(2)-16*x^3-8*x^2-160*x)*log(x)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+2 5)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="maxima")
Output:
-2*(8*x^2 - (x^3 + x^2 + 2*x*(log(2) + 5))*log(x))/((log(2) + 5)*log(x) - 4*x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.33 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=\frac {2 \, x^{3}}{\log \left (2\right ) + 5} + 4 \, x + \frac {2 \, x^{2}}{\log \left (2\right ) + 5} + \frac {8 \, {\left (x^{4} + x^{3}\right )}}{\log \left (2\right )^{2} \log \left (x\right ) - 4 \, x \log \left (2\right ) + 10 \, \log \left (2\right ) \log \left (x\right ) - 20 \, x + 25 \, \log \left (x\right )} \] Input:
integrate(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-3 2*x*log(2)-16*x^3-8*x^2-160*x)*log(x)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+2 5)*log(x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x, algorithm="giac")
Output:
2*x^3/(log(2) + 5) + 4*x + 2*x^2/(log(2) + 5) + 8*(x^4 + x^3)/(log(2)^2*lo g(x) - 4*x*log(2) + 10*log(2)*log(x) - 20*x + 25*log(x))
Timed out. \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=-\int \frac {\ln \left (x\right )\,\left (160\,x+32\,x\,\ln \left (2\right )+8\,x^2+16\,x^3\right )-{\ln \left (x\right )}^2\,\left (20\,x+\ln \left (2\right )\,\left (6\,x^2+4\,x+40\right )+4\,{\ln \left (2\right )}^2+30\,x^2+100\right )-56\,x^2+8\,x^3}{{\ln \left (x\right )}^2\,\left (10\,\ln \left (2\right )+{\ln \left (2\right )}^2+25\right )-\ln \left (x\right )\,\left (40\,x+8\,x\,\ln \left (2\right )\right )+16\,x^2} \,d x \] Input:
int(-(log(x)*(160*x + 32*x*log(2) + 8*x^2 + 16*x^3) - log(x)^2*(20*x + log (2)*(4*x + 6*x^2 + 40) + 4*log(2)^2 + 30*x^2 + 100) - 56*x^2 + 8*x^3)/(log (x)^2*(10*log(2) + log(2)^2 + 25) - log(x)*(40*x + 8*x*log(2)) + 16*x^2),x )
Output:
-int((log(x)*(160*x + 32*x*log(2) + 8*x^2 + 16*x^3) - log(x)^2*(20*x + log (2)*(4*x + 6*x^2 + 40) + 4*log(2)^2 + 30*x^2 + 100) - 56*x^2 + 8*x^3)/(log (x)^2*(10*log(2) + log(2)^2 + 25) - log(x)*(40*x + 8*x*log(2)) + 16*x^2), x)
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {56 x^2-8 x^3+\left (-160 x-8 x^2-16 x^3-32 x \log (2)\right ) \log (x)+\left (100+20 x+30 x^2+\left (40+4 x+6 x^2\right ) \log (2)+4 \log ^2(2)\right ) \log ^2(x)}{16 x^2+(-40 x-8 x \log (2)) \log (x)+\left (25+10 \log (2)+\log ^2(2)\right ) \log ^2(x)} \, dx=\frac {2 x \left (2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+\mathrm {log}\left (x \right ) x^{2}+\mathrm {log}\left (x \right ) x +10 \,\mathrm {log}\left (x \right )-8 x \right )}{\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+5 \,\mathrm {log}\left (x \right )-4 x} \] Input:
int(((4*log(2)^2+(6*x^2+4*x+40)*log(2)+30*x^2+20*x+100)*log(x)^2+(-32*x*lo g(2)-16*x^3-8*x^2-160*x)*log(x)-8*x^3+56*x^2)/((log(2)^2+10*log(2)+25)*log (x)^2+(-8*x*log(2)-40*x)*log(x)+16*x^2),x)
Output:
(2*x*(2*log(x)*log(2) + log(x)*x**2 + log(x)*x + 10*log(x) - 8*x))/(log(x) *log(2) + 5*log(x) - 4*x)