Integrand size = 89, antiderivative size = 24 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=x \left (3+x+x^2 (5+x) \left (2+\frac {3}{2+x}+\log (\log (x))\right )\right ) \] Output:
x*(3+x^2*(5+x)*(2+ln(ln(x))+3/(2+x))+x)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=-15 x+10 x^2+13 x^3+2 x^4-\frac {72}{2+x}+x^3 (5+x) \log (\log (x)) \] Input:
Integrate[(20*x^2 + 24*x^3 + 9*x^4 + x^5 + (12 + 20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5)*Log[x] + (60*x^2 + 76*x^3 + 31*x^4 + 4*x^5)*Log[x]*Log[L og[x]])/((4 + 4*x + x^2)*Log[x]),x]
Output:
-15*x + 10*x^2 + 13*x^3 + 2*x^4 - 72/(2 + x) + x^3*(5 + 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.
\(\displaystyle \int \frac {x^5+9 x^4+24 x^3+20 x^2+\left (8 x^5+71 x^4+208 x^3+221 x^2+20 x+12\right ) \log (x)+\left (4 x^5+31 x^4+76 x^3+60 x^2\right ) \log (x) \log (\log (x))}{\left (x^2+4 x+4\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {x^5+9 x^4+24 x^3+20 x^2+\left (8 x^5+71 x^4+208 x^3+221 x^2+20 x+12\right ) \log (x)+\left (4 x^5+31 x^4+76 x^3+60 x^2\right ) \log (x) \log (\log (x))}{(x+2)^2 \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^5}{(x+2)^2 \log (x)}+\frac {9 x^4}{(x+2)^2 \log (x)}+\frac {24 x^3}{(x+2)^2 \log (x)}+(4 x+15) x^2 \log (\log (x))+\frac {20 x^2}{(x+2)^2 \log (x)}+\frac {8 x^5+71 x^4+208 x^3+221 x^2+20 x+12}{(x+2)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {x^5}{(x+2)^2 \log (x)}dx+9 \int \frac {x^4}{(x+2)^2 \log (x)}dx+24 \int \frac {x^3}{(x+2)^2 \log (x)}dx+20 \int \frac {x^2}{(x+2)^2 \log (x)}dx-5 \operatorname {ExpIntegralEi}(3 \log (x))-\operatorname {ExpIntegralEi}(4 \log (x))+2 x^4+x^4 \log (\log (x))+13 x^3+5 x^3 \log (\log (x))+10 x^2-15 x-\frac {72}{x+2}\) |
Input:
Int[(20*x^2 + 24*x^3 + 9*x^4 + x^5 + (12 + 20*x + 221*x^2 + 208*x^3 + 71*x ^4 + 8*x^5)*Log[x] + (60*x^2 + 76*x^3 + 31*x^4 + 4*x^5)*Log[x]*Log[Log[x]] )/((4 + 4*x + x^2)*Log[x]),x]
Output:
$Aborted
Time = 0.37 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75
method | result | size |
default | \(2 x^{4}+13 x^{3}+10 x^{2}-15 x -\frac {72}{2+x}+5 \ln \left (\ln \left (x \right )\right ) x^{3}+\ln \left (\ln \left (x \right )\right ) x^{4}\) | \(42\) |
parts | \(2 x^{4}+13 x^{3}+10 x^{2}-15 x -\frac {72}{2+x}+5 \ln \left (\ln \left (x \right )\right ) x^{3}+\ln \left (\ln \left (x \right )\right ) x^{4}\) | \(42\) |
risch | \(\left (x^{4}+5 x^{3}\right ) \ln \left (\ln \left (x \right )\right )+\frac {2 x^{5}+17 x^{4}+36 x^{3}+5 x^{2}-30 x -72}{2+x}\) | \(46\) |
parallelrisch | \(-\frac {-2 x^{5} \ln \left (\ln \left (x \right )\right )-4 x^{5}-14 \ln \left (\ln \left (x \right )\right ) x^{4}-34 x^{4}-20 \ln \left (\ln \left (x \right )\right ) x^{3}-72 x^{3}-10 x^{2}+24}{2 \left (2+x \right )}\) | \(54\) |
Input:
int(((4*x^5+31*x^4+76*x^3+60*x^2)*ln(x)*ln(ln(x))+(8*x^5+71*x^4+208*x^3+22 1*x^2+20*x+12)*ln(x)+x^5+9*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/ln(x),x,method=_ RETURNVERBOSE)
Output:
2*x^4+13*x^3+10*x^2-15*x-72/(2+x)+5*ln(ln(x))*x^3+ln(ln(x))*x^4
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=\frac {2 \, x^{5} + 17 \, x^{4} + 36 \, x^{3} + 5 \, x^{2} + {\left (x^{5} + 7 \, x^{4} + 10 \, x^{3}\right )} \log \left (\log \left (x\right )\right ) - 30 \, x - 72}{x + 2} \] Input:
integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+2 08*x^3+221*x^2+20*x+12)*log(x)+x^5+9*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x) ,x, algorithm="fricas")
Output:
(2*x^5 + 17*x^4 + 36*x^3 + 5*x^2 + (x^5 + 7*x^4 + 10*x^3)*log(log(x)) - 30 *x - 72)/(x + 2)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=2 x^{4} + 13 x^{3} + 10 x^{2} - 15 x + \left (x^{4} + 5 x^{3}\right ) \log {\left (\log {\left (x \right )} \right )} - \frac {72}{x + 2} \] Input:
integrate(((4*x**5+31*x**4+76*x**3+60*x**2)*ln(x)*ln(ln(x))+(8*x**5+71*x** 4+208*x**3+221*x**2+20*x+12)*ln(x)+x**5+9*x**4+24*x**3+20*x**2)/(x**2+4*x+ 4)/ln(x),x)
Output:
2*x**4 + 13*x**3 + 10*x**2 - 15*x + (x**4 + 5*x**3)*log(log(x)) - 72/(x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=\frac {2 \, x^{5} + 17 \, x^{4} + 36 \, x^{3} + 5 \, x^{2} + {\left (x^{5} + 7 \, x^{4} + 10 \, x^{3}\right )} \log \left (\log \left (x\right )\right ) - 30 \, x - 72}{x + 2} \] Input:
integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+2 08*x^3+221*x^2+20*x+12)*log(x)+x^5+9*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x) ,x, algorithm="maxima")
Output:
(2*x^5 + 17*x^4 + 36*x^3 + 5*x^2 + (x^5 + 7*x^4 + 10*x^3)*log(log(x)) - 30 *x - 72)/(x + 2)
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=2 \, x^{4} + 13 \, x^{3} + 10 \, x^{2} + {\left (x^{4} + 5 \, x^{3}\right )} \log \left (\log \left (x\right )\right ) - 15 \, x - \frac {72}{x + 2} \] Input:
integrate(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+2 08*x^3+221*x^2+20*x+12)*log(x)+x^5+9*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x) ,x, algorithm="giac")
Output:
2*x^4 + 13*x^3 + 10*x^2 + (x^4 + 5*x^3)*log(log(x)) - 15*x - 72/(x + 2)
Time = 3.58 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=\ln \left (\ln \left (x\right )\right )\,\left (x^4+5\,x^3\right )-15\,x-\frac {72}{x+2}+10\,x^2+13\,x^3+2\,x^4 \] Input:
int((log(x)*(20*x + 221*x^2 + 208*x^3 + 71*x^4 + 8*x^5 + 12) + 20*x^2 + 24 *x^3 + 9*x^4 + x^5 + log(log(x))*log(x)*(60*x^2 + 76*x^3 + 31*x^4 + 4*x^5) )/(log(x)*(4*x + x^2 + 4)),x)
Output:
log(log(x))*(5*x^3 + x^4) - 15*x - 72/(x + 2) + 10*x^2 + 13*x^3 + 2*x^4
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {20 x^2+24 x^3+9 x^4+x^5+\left (12+20 x+221 x^2+208 x^3+71 x^4+8 x^5\right ) \log (x)+\left (60 x^2+76 x^3+31 x^4+4 x^5\right ) \log (x) \log (\log (x))}{\left (4+4 x+x^2\right ) \log (x)} \, dx=\frac {x \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{4}+7 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{3}+10 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+2 x^{4}+17 x^{3}+36 x^{2}+5 x +6\right )}{x +2} \] Input:
int(((4*x^5+31*x^4+76*x^3+60*x^2)*log(x)*log(log(x))+(8*x^5+71*x^4+208*x^3 +221*x^2+20*x+12)*log(x)+x^5+9*x^4+24*x^3+20*x^2)/(x^2+4*x+4)/log(x),x)
Output:
(x*(log(log(x))*x**4 + 7*log(log(x))*x**3 + 10*log(log(x))*x**2 + 2*x**4 + 17*x**3 + 36*x**2 + 5*x + 6))/(x + 2)