Integrand size = 106, antiderivative size = 25 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (\frac {x}{5 (-4-x)}\right )\right )} \] Output:
exp(1/5*x^2*ln(ln(1/5*x/(-4-x)))*ln(x))
Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=x^{\frac {1}{5} x^2 \log \left (\log \left (-\frac {x}{5 (4+x)}\right )\right )} \] Input:
Integrate[(E^((x^2*Log[x]*Log[Log[-(x/(20 + 5*x))]])/5)*(4*x*Log[x] + ((4* x + x^2)*Log[-(x/(20 + 5*x))] + (8*x + 2*x^2)*Log[x]*Log[-(x/(20 + 5*x))]) *Log[Log[-(x/(20 + 5*x))]]))/((20 + 5*x)*Log[-(x/(20 + 5*x))]),x]
Output:
x^((x^2*Log[Log[-1/5*x/(4 + x)]])/5)
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.
\(\displaystyle \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} \left (\left (\left (x^2+4 x\right ) \log \left (-\frac {x}{5 x+20}\right )+\left (2 x^2+8 x\right ) \log (x) \log \left (-\frac {x}{5 x+20}\right )\right ) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )+4 x \log (x)\right )}{(5 x+20) \log \left (-\frac {x}{5 x+20}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} \log (x)}{5 (x+4) \log \left (-\frac {x}{5 (x+4)}\right )}+\frac {1}{5} x e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} (2 \log (x)+1) \log \left (\log \left (-\frac {x}{5 (x+4)}\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{5} \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} \log (x)}{\log \left (-\frac {x}{5 (x+4)}\right )}dx-\frac {16}{5} \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} \log (x)}{(x+4) \log \left (-\frac {x}{5 (x+4)}\right )}dx+\frac {1}{5} \int e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} x \log \left (\log \left (-\frac {x}{5 (x+4)}\right )\right )dx+\frac {2}{5} \int e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{5 x+20}\right )\right )} x \log (x) \log \left (\log \left (-\frac {x}{5 (x+4)}\right )\right )dx\) |
Input:
Int[(E^((x^2*Log[x]*Log[Log[-(x/(20 + 5*x))]])/5)*(4*x*Log[x] + ((4*x + x^ 2)*Log[-(x/(20 + 5*x))] + (8*x + 2*x^2)*Log[x]*Log[-(x/(20 + 5*x))])*Log[L og[-(x/(20 + 5*x))]]))/((20 + 5*x)*Log[-(x/(20 + 5*x))]),x]
Output:
$Aborted
Time = 9.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {x^{2} \ln \left (x \right ) \ln \left (\ln \left (-\frac {x}{5 \left (4+x \right )}\right )\right )}{5}}\) | \(19\) |
risch | \(\left (-\ln \left (5\right )+i \pi +\ln \left (x \right )-\ln \left (4+x \right )+\frac {i \pi \,\operatorname {csgn}\left (\frac {i x}{4+x}\right ) \left (-\operatorname {csgn}\left (\frac {i x}{4+x}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (\operatorname {csgn}\left (\frac {i x}{4+x}\right )-\operatorname {csgn}\left (\frac {i}{4+x}\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i x}{4+x}\right )^{2} \left (\operatorname {csgn}\left (\frac {i x}{4+x}\right )-1\right )\right )^{\frac {x^{2} \ln \left (x \right )}{5}}\) | \(108\) |
Input:
int((((2*x^2+8*x)*ln(-x/(20+5*x))*ln(x)+(x^2+4*x)*ln(-x/(20+5*x)))*ln(ln(- x/(20+5*x)))+4*x*ln(x))*exp(1/5*x^2*ln(x)*ln(ln(-x/(20+5*x))))/(20+5*x)/ln (-x/(20+5*x)),x,method=_RETURNVERBOSE)
Output:
exp(1/5*x^2*ln(x)*ln(ln(-1/5*x/(4+x))))
Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=e^{\left (\frac {1}{5} \, x^{2} \log \left (x\right ) \log \left (\log \left (-\frac {x}{5 \, {\left (x + 4\right )}}\right )\right )\right )} \] Input:
integrate((((2*x^2+8*x)*log(-x/(20+5*x))*log(x)+(x^2+4*x)*log(-x/(20+5*x)) )*log(log(-x/(20+5*x)))+4*x*log(x))*exp(1/5*x^2*log(x)*log(log(-x/(20+5*x) )))/(20+5*x)/log(-x/(20+5*x)),x, algorithm="fricas")
Output:
e^(1/5*x^2*log(x)*log(log(-1/5*x/(x + 4))))
Time = 1.75 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=e^{\frac {x^{2} \log {\left (x \right )} \log {\left (\log {\left (- \frac {x}{5 x + 20} \right )} \right )}}{5}} \] Input:
integrate((((2*x**2+8*x)*ln(-x/(20+5*x))*ln(x)+(x**2+4*x)*ln(-x/(20+5*x))) *ln(ln(-x/(20+5*x)))+4*x*ln(x))*exp(1/5*x**2*ln(x)*ln(ln(-x/(20+5*x))))/(2 0+5*x)/ln(-x/(20+5*x)),x)
Output:
exp(x**2*log(x)*log(log(-x/(5*x + 20)))/5)
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=e^{\left (\frac {1}{5} \, x^{2} \log \left (x\right ) \log \left (-\log \left (5\right ) + \log \left (x\right ) - \log \left (-x - 4\right )\right )\right )} \] Input:
integrate((((2*x^2+8*x)*log(-x/(20+5*x))*log(x)+(x^2+4*x)*log(-x/(20+5*x)) )*log(log(-x/(20+5*x)))+4*x*log(x))*exp(1/5*x^2*log(x)*log(log(-x/(20+5*x) )))/(20+5*x)/log(-x/(20+5*x)),x, algorithm="maxima")
Output:
e^(1/5*x^2*log(x)*log(-log(5) + log(x) - log(-x - 4)))
Time = 0.16 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=e^{\left (\frac {1}{5} \, x^{2} \log \left (x\right ) \log \left (\log \left (-\frac {x}{5 \, {\left (x + 4\right )}}\right )\right )\right )} \] Input:
integrate((((2*x^2+8*x)*log(-x/(20+5*x))*log(x)+(x^2+4*x)*log(-x/(20+5*x)) )*log(log(-x/(20+5*x)))+4*x*log(x))*exp(1/5*x^2*log(x)*log(log(-x/(20+5*x) )))/(20+5*x)/log(-x/(20+5*x)),x, algorithm="giac")
Output:
e^(1/5*x^2*log(x)*log(log(-1/5*x/(x + 4))))
Time = 2.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx={\mathrm {e}}^{\frac {x^2\,\ln \left (\ln \left (-\frac {x}{5\,x+20}\right )\right )\,\ln \left (x\right )}{5}} \] Input:
int((exp((x^2*log(log(-x/(5*x + 20)))*log(x))/5)*(4*x*log(x) + log(log(-x/ (5*x + 20)))*(log(-x/(5*x + 20))*(4*x + x^2) + log(-x/(5*x + 20))*log(x)*( 8*x + 2*x^2))))/(log(-x/(5*x + 20))*(5*x + 20)),x)
Output:
exp((x^2*log(log(-x/(5*x + 20)))*log(x))/5)
Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{\frac {1}{5} x^2 \log (x) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )} \left (4 x \log (x)+\left (\left (4 x+x^2\right ) \log \left (-\frac {x}{20+5 x}\right )+\left (8 x+2 x^2\right ) \log (x) \log \left (-\frac {x}{20+5 x}\right )\right ) \log \left (\log \left (-\frac {x}{20+5 x}\right )\right )\right )}{(20+5 x) \log \left (-\frac {x}{20+5 x}\right )} \, dx=\mathrm {log}\left (-\frac {x}{5 x +20}\right )^{\frac {\mathrm {log}\left (x \right ) x^{2}}{5}} \] Input:
int((((2*x^2+8*x)*log(-x/(20+5*x))*log(x)+(x^2+4*x)*log(-x/(20+5*x)))*log( log(-x/(20+5*x)))+4*x*log(x))*exp(1/5*x^2*log(x)*log(log(-x/(20+5*x))))/(2 0+5*x)/log(-x/(20+5*x)),x)
Output:
log(( - x)/(5*x + 20))**((log(x)*x**2)/5)