Integrand size = 158, antiderivative size = 24 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=x+\frac {\log (\log (4))}{2 x+\frac {x}{1+2 x}+\log (\log (x))} \] Output:
x-ln(1/2/ln(2))/(ln(ln(x))+x/(1+2*x)+2*x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=x+\frac {(1+2 x) \log (\log (4))}{3 x+4 x^2+\log (\log (x))+2 x \log (\log (x))} \] Input:
Integrate[((9*x^3 + 24*x^4 + 16*x^5)*Log[x] + (-1 - 4*x - 4*x^2 + (-3*x - 8*x^2 - 8*x^3)*Log[x])*Log[Log[4]] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[ Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2)/((9*x^3 + 24*x^4 + 16* x^5)*Log[x] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2),x]
Output:
x + ((1 + 2*x)*Log[Log[4]])/(3*x + 4*x^2 + Log[Log[x]] + 2*x*Log[Log[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.
| \(\displaystyle \int \frac {\left (4 x^3+4 x^2+x\right ) \log (x) \log ^2(\log (x))+\log (\log (4)) \left (-4 x^2+\left (-8 x^3-8 x^2-3 x\right ) \log (x)-4 x-1\right )+\left (16 x^5+24 x^4+9 x^3\right ) \log (x)+\left (16 x^4+20 x^3+6 x^2\right ) \log (x) \log (\log (x))}{\left (4 x^3+4 x^2+x\right ) \log (x) \log ^2(\log (x))+\left (16 x^5+24 x^4+9 x^3\right ) \log (x)+\left (16 x^4+20 x^3+6 x^2\right ) \log (x) \log (\log (x))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (4 x^3+4 x^2+x\right ) \log (x) \log ^2(\log (x))+\log (\log (4)) \left (-4 x^2+\left (-8 x^3-8 x^2-3 x\right ) \log (x)-4 x-1\right )+\left (16 x^5+24 x^4+9 x^3\right ) \log (x)+\left (16 x^4+20 x^3+6 x^2\right ) \log (x) \log (\log (x))}{x \log (x) \left (4 x^2+3 x+2 x \log (\log (x))+\log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (1-\frac {\log (\log (4)) \left (8 x^3 \log (x)+4 x^2+8 x^2 \log (x)+4 x+3 x \log (x)+1\right )}{x \log (x) \left (4 x^2+3 x+2 x \log (\log (x))+\log (\log (x))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \log (\log (4)) \int \frac {1}{\left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx-8 \log (\log (4)) \int \frac {x}{\left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx-8 \log (\log (4)) \int \frac {x^2}{\left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx-4 \log (\log (4)) \int \frac {1}{\log (x) \left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx-\log (\log (4)) \int \frac {1}{x \log (x) \left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx-4 \log (\log (4)) \int \frac {x}{\log (x) \left (4 x^2+2 \log (\log (x)) x+3 x+\log (\log (x))\right )^2}dx+x\) |
Input:
Int[((9*x^3 + 24*x^4 + 16*x^5)*Log[x] + (-1 - 4*x - 4*x^2 + (-3*x - 8*x^2 - 8*x^3)*Log[x])*Log[Log[4]] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x] ] + (x + 4*x^2 + 4*x^3)*Log[x]*Log[Log[x]]^2)/((9*x^3 + 24*x^4 + 16*x^5)*L og[x] + (6*x^2 + 20*x^3 + 16*x^4)*Log[x]*Log[Log[x]] + (x + 4*x^2 + 4*x^3) *Log[x]*Log[Log[x]]^2),x]
Output:
$Aborted
Time = 1.80 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
| method | result | size |
| risch | \(x +\frac {\left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right ) \left (1+2 x \right )}{2 x \ln \left (\ln \left (x \right )\right )+4 x^{2}+\ln \left (\ln \left (x \right )\right )+3 x}\) | \(35\) |
| parallelrisch | \(\frac {-3 x +4 x^{2} \ln \left (\ln \left (x \right )\right )-\ln \left (\ln \left (x \right )\right )+2 \ln \left (2 \ln \left (2\right )\right )+2 x^{2}+8 x^{3}+4 \ln \left (2 \ln \left (2\right )\right ) x}{4 x \ln \left (\ln \left (x \right )\right )+8 x^{2}+2 \ln \left (\ln \left (x \right )\right )+6 x}\) | \(65\) |
Input:
int(((4*x^3+4*x^2+x)*ln(x)*ln(ln(x))^2+(16*x^4+20*x^3+6*x^2)*ln(x)*ln(ln(x ))+((-8*x^3-8*x^2-3*x)*ln(x)-4*x^2-4*x-1)*ln(2*ln(2))+(16*x^5+24*x^4+9*x^3 )*ln(x))/((4*x^3+4*x^2+x)*ln(x)*ln(ln(x))^2+(16*x^4+20*x^3+6*x^2)*ln(x)*ln (ln(x))+(16*x^5+24*x^4+9*x^3)*ln(x)),x,method=_RETURNVERBOSE)
Output:
x+(ln(2)+ln(ln(2)))*(1+2*x)/(2*x*ln(ln(x))+4*x^2+ln(ln(x))+3*x)
Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.25 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {4 \, x^{3} + 3 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (2 \, \log \left (2\right )\right ) + {\left (2 \, x^{2} + x\right )} \log \left (\log \left (x\right )\right )}{4 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) + 3 \, x} \] Input:
integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log( x)*log(log(x))+((-8*x^3-8*x^2-3*x)*log(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x ^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20* x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm=" fricas")
Output:
(4*x^3 + 3*x^2 + (2*x + 1)*log(2*log(2)) + (2*x^2 + x)*log(log(x)))/(4*x^2 + (2*x + 1)*log(log(x)) + 3*x)
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=x + \frac {2 x \log {\left (\log {\left (2 \right )} \right )} + 2 x \log {\left (2 \right )} + \log {\left (\log {\left (2 \right )} \right )} + \log {\left (2 \right )}}{4 x^{2} + 3 x + \left (2 x + 1\right ) \log {\left (\log {\left (x \right )} \right )}} \] Input:
integrate(((4*x**3+4*x**2+x)*ln(x)*ln(ln(x))**2+(16*x**4+20*x**3+6*x**2)*l n(x)*ln(ln(x))+((-8*x**3-8*x**2-3*x)*ln(x)-4*x**2-4*x-1)*ln(2*ln(2))+(16*x **5+24*x**4+9*x**3)*ln(x))/((4*x**3+4*x**2+x)*ln(x)*ln(ln(x))**2+(16*x**4+ 20*x**3+6*x**2)*ln(x)*ln(ln(x))+(16*x**5+24*x**4+9*x**3)*ln(x)),x)
Output:
x + (2*x*log(log(2)) + 2*x*log(2) + log(log(2)) + log(2))/(4*x**2 + 3*x + (2*x + 1)*log(log(x)))
Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {4 \, x^{3} + 3 \, x^{2} + 2 \, x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + {\left (2 \, x^{2} + x\right )} \log \left (\log \left (x\right )\right ) + \log \left (2\right ) + \log \left (\log \left (2\right )\right )}{4 \, x^{2} + {\left (2 \, x + 1\right )} \log \left (\log \left (x\right )\right ) + 3 \, x} \] Input:
integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log( x)*log(log(x))+((-8*x^3-8*x^2-3*x)*log(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x ^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20* x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm=" maxima")
Output:
(4*x^3 + 3*x^2 + 2*x*(log(2) + log(log(2))) + (2*x^2 + x)*log(log(x)) + lo g(2) + log(log(2)))/(4*x^2 + (2*x + 1)*log(log(x)) + 3*x)
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=x + \frac {2 \, x \log \left (2\right ) + 2 \, x \log \left (\log \left (2\right )\right ) + \log \left (2\right ) + \log \left (\log \left (2\right )\right )}{4 \, x^{2} + 2 \, x \log \left (\log \left (x\right )\right ) + 3 \, x + \log \left (\log \left (x\right )\right )} \] Input:
integrate(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log( x)*log(log(x))+((-8*x^3-8*x^2-3*x)*log(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x ^5+24*x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20* x^3+6*x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x, algorithm=" giac")
Output:
x + (2*x*log(2) + 2*x*log(log(2)) + log(2) + log(log(2)))/(4*x^2 + 2*x*log (log(x)) + 3*x + log(log(x)))
Timed out. \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=\int \frac {\ln \left (x\right )\,\left (4\,x^3+4\,x^2+x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\ln \left (x\right )\,\left (16\,x^4+20\,x^3+6\,x^2\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left (16\,x^5+24\,x^4+9\,x^3\right )-\ln \left (2\,\ln \left (2\right )\right )\,\left (4\,x+4\,x^2+\ln \left (x\right )\,\left (8\,x^3+8\,x^2+3\,x\right )+1\right )}{\ln \left (x\right )\,\left (4\,x^3+4\,x^2+x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\ln \left (x\right )\,\left (16\,x^4+20\,x^3+6\,x^2\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left (16\,x^5+24\,x^4+9\,x^3\right )} \,d x \] Input:
int((log(x)*(9*x^3 + 24*x^4 + 16*x^5) - log(2*log(2))*(4*x + 4*x^2 + log(x )*(3*x + 8*x^2 + 8*x^3) + 1) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4))/(log(x)*(9*x^3 + 24*x^4 + 16 *x^5) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x ^2 + 20*x^3 + 16*x^4)),x)
Output:
int((log(x)*(9*x^3 + 24*x^4 + 16*x^5) - log(2*log(2))*(4*x + 4*x^2 + log(x )*(3*x + 8*x^2 + 8*x^3) + 1) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x^2 + 20*x^3 + 16*x^4))/(log(x)*(9*x^3 + 24*x^4 + 16 *x^5) + log(log(x))^2*log(x)*(x + 4*x^2 + 4*x^3) + log(log(x))*log(x)*(6*x ^2 + 20*x^3 + 16*x^4)), x)
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (-1-4 x-4 x^2+\left (-3 x-8 x^2-8 x^3\right ) \log (x)\right ) \log (\log (4))+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))}{\left (9 x^3+24 x^4+16 x^5\right ) \log (x)+\left (6 x^2+20 x^3+16 x^4\right ) \log (x) \log (\log (x))+\left (x+4 x^2+4 x^3\right ) \log (x) \log ^2(\log (x))} \, dx=\frac {2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +2 \,\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right ) x +\mathrm {log}\left (2 \,\mathrm {log}\left (2\right )\right )+4 x^{3}+3 x^{2}}{2 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x +\mathrm {log}\left (\mathrm {log}\left (x \right )\right )+4 x^{2}+3 x} \] Input:
int(((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6*x^2)*log(x)*log (log(x))+((-8*x^3-8*x^2-3*x)*log(x)-4*x^2-4*x-1)*log(2*log(2))+(16*x^5+24* x^4+9*x^3)*log(x))/((4*x^3+4*x^2+x)*log(x)*log(log(x))^2+(16*x^4+20*x^3+6* x^2)*log(x)*log(log(x))+(16*x^5+24*x^4+9*x^3)*log(x)),x)
Output:
(2*log(log(x))*x**2 + log(log(x))*x + 2*log(2*log(2))*x + log(2*log(2)) + 4*x**3 + 3*x**2)/(2*log(log(x))*x + log(log(x)) + 4*x**2 + 3*x)