Integrand size = 93, antiderivative size = 22 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=\log ^2(x)-\log \left (\frac {1}{9} x \log (4+\log (x (5+x)))\right ) \] Output:
ln(x)^2-ln(1/9*x*ln(4+ln((5+x)*x)))
Time = 0.13 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=-\log (x)+\log ^2(x)-\log (\log (4+\log (x (5+x)))) \] Input:
Integrate[(-5 - 2*x + (-20 - 4*x + (40 + 8*x)*Log[x] + (-5 - x + (10 + 2*x )*Log[x])*Log[5*x + x^2])*Log[4 + Log[5*x + x^2]])/((20*x + 4*x^2 + (5*x + x^2)*Log[5*x + x^2])*Log[4 + Log[5*x + x^2]]),x]
Output:
-Log[x] + Log[x]^2 - Log[Log[4 + Log[x*(5 + x)]]]
Time = 1.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {7292, 7293, 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.
\(\displaystyle \int \frac {\left ((-x+(2 x+10) \log (x)-5) \log \left (x^2+5 x\right )-4 x+(8 x+40) \log (x)-20\right ) \log \left (\log \left (x^2+5 x\right )+4\right )-2 x-5}{\left (4 x^2+\left (x^2+5 x\right ) \log \left (x^2+5 x\right )+20 x\right ) \log \left (\log \left (x^2+5 x\right )+4\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left ((-x+(2 x+10) \log (x)-5) \log \left (x^2+5 x\right )-4 x+(8 x+40) \log (x)-20\right ) \log \left (\log \left (x^2+5 x\right )+4\right )-2 x-5}{x (x+5) (\log (x (x+5))+4) \log \left (\log \left (x^2+5 x\right )+4\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-2 x-5}{x (x+5) (\log (x (x+5))+4) \log (\log (x (x+5))+4)}+\frac {2 \log (x)-1}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} (1-2 \log (x))^2-\log (\log (\log (x (x+5))+4))\) |
Input:
Int[(-5 - 2*x + (-20 - 4*x + (40 + 8*x)*Log[x] + (-5 - x + (10 + 2*x)*Log[ x])*Log[5*x + x^2])*Log[4 + Log[5*x + x^2]])/((20*x + 4*x^2 + (5*x + x^2)* Log[5*x + x^2])*Log[4 + Log[5*x + x^2]]),x]
Output:
(1 - 2*Log[x])^2/4 - Log[Log[4 + Log[x*(5 + x)]]]
Time = 1.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x \right )^{2}-\ln \left (\ln \left (\ln \left (x^{2}+5 x \right )+4\right )\right )\) | \(24\) |
parallelrisch | \(-\ln \left (x \right )+\ln \left (x \right )^{2}-\ln \left (\ln \left (\ln \left (x^{2}+5 x \right )+4\right )\right )\) | \(24\) |
parts | \(-\ln \left (x \right )+\ln \left (x \right )^{2}-\ln \left (\ln \left (\ln \left (x^{2}+5 x \right )+4\right )\right )\) | \(24\) |
risch | \(\ln \left (x \right )^{2}-\ln \left (x \right )-\ln \left (\ln \left (\ln \left (x \right )+\ln \left (5+x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x \left (5+x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (5+x \right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i x \left (5+x \right )\right )+\operatorname {csgn}\left (i \left (5+x \right )\right )\right )}{2}+4\right )\right )\) | \(68\) |
Input:
int(((((2*x+10)*ln(x)-x-5)*ln(x^2+5*x)+(8*x+40)*ln(x)-4*x-20)*ln(ln(x^2+5* x)+4)-2*x-5)/((x^2+5*x)*ln(x^2+5*x)+4*x^2+20*x)/ln(ln(x^2+5*x)+4),x,method =_RETURNVERBOSE)
Output:
-ln(x)+ln(x)^2-ln(ln(ln(x^2+5*x)+4))
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=\log \left (x\right )^{2} - \log \left (x\right ) - \log \left (\log \left (\log \left (x^{2} + 5 \, x\right ) + 4\right )\right ) \] Input:
integrate(((((2*x+10)*log(x)-x-5)*log(x^2+5*x)+(8*x+40)*log(x)-4*x-20)*log (log(x^2+5*x)+4)-2*x-5)/((x^2+5*x)*log(x^2+5*x)+4*x^2+20*x)/log(log(x^2+5* x)+4),x, algorithm="fricas")
Output:
log(x)^2 - log(x) - log(log(log(x^2 + 5*x) + 4))
Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=\log {\left (x \right )}^{2} - \log {\left (x \right )} - \log {\left (\log {\left (\log {\left (x^{2} + 5 x \right )} + 4 \right )} \right )} \] Input:
integrate(((((2*x+10)*ln(x)-x-5)*ln(x**2+5*x)+(8*x+40)*ln(x)-4*x-20)*ln(ln (x**2+5*x)+4)-2*x-5)/((x**2+5*x)*ln(x**2+5*x)+4*x**2+20*x)/ln(ln(x**2+5*x) +4),x)
Output:
log(x)**2 - log(x) - log(log(log(x**2 + 5*x) + 4))
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=\log \left (x\right )^{2} - \log \left (x\right ) - \log \left (\log \left (\log \left (x + 5\right ) + \log \left (x\right ) + 4\right )\right ) \] Input:
integrate(((((2*x+10)*log(x)-x-5)*log(x^2+5*x)+(8*x+40)*log(x)-4*x-20)*log (log(x^2+5*x)+4)-2*x-5)/((x^2+5*x)*log(x^2+5*x)+4*x^2+20*x)/log(log(x^2+5* x)+4),x, algorithm="maxima")
Output:
log(x)^2 - log(x) - log(log(log(x + 5) + log(x) + 4))
Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=\log \left (x\right )^{2} - \log \left (x\right ) - \log \left ({\left | \log \left (\log \left (x + 5\right ) + \log \left (x\right ) + 4\right ) \right |}\right ) \] Input:
integrate(((((2*x+10)*log(x)-x-5)*log(x^2+5*x)+(8*x+40)*log(x)-4*x-20)*log (log(x^2+5*x)+4)-2*x-5)/((x^2+5*x)*log(x^2+5*x)+4*x^2+20*x)/log(log(x^2+5* x)+4),x, algorithm="giac")
Output:
log(x)^2 - log(x) - log(abs(log(log(x + 5) + log(x) + 4)))
Time = 3.93 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx={\ln \left (x\right )}^2-\ln \left (x\right )-\ln \left (\ln \left (\ln \left (x^2+5\,x\right )+4\right )\right ) \] Input:
int(-(2*x + log(log(5*x + x^2) + 4)*(4*x + log(5*x + x^2)*(x - log(x)*(2*x + 10) + 5) - log(x)*(8*x + 40) + 20) + 5)/(log(log(5*x + x^2) + 4)*(20*x + 4*x^2 + log(5*x + x^2)*(5*x + x^2))),x)
Output:
log(x)^2 - log(x) - log(log(log(5*x + x^2) + 4))
Time = 0.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {-5-2 x+\left (-20-4 x+(40+8 x) \log (x)+(-5-x+(10+2 x) \log (x)) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )}{\left (20 x+4 x^2+\left (5 x+x^2\right ) \log \left (5 x+x^2\right )\right ) \log \left (4+\log \left (5 x+x^2\right )\right )} \, dx=-\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (x^{2}+5 x \right )+4\right )\right )+\mathrm {log}\left (x \right )^{2}-\mathrm {log}\left (x \right ) \] Input:
int(((((2*x+10)*log(x)-x-5)*log(x^2+5*x)+(8*x+40)*log(x)-4*x-20)*log(log(x ^2+5*x)+4)-2*x-5)/((x^2+5*x)*log(x^2+5*x)+4*x^2+20*x)/log(log(x^2+5*x)+4), x)
Output:
- log(log(log(x**2 + 5*x) + 4)) + log(x)**2 - log(x)