Integrand size = 259, antiderivative size = 33 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=-x+\frac {x}{-5+\frac {-x+\log \left (e^5+\frac {x}{4}\right )}{-8-\log (x)}} \] Output:
x/((ln(exp(5)+1/4*x)-x)/(-8-ln(x))-5)-x
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=-\frac {1}{5} x \left (6+\frac {x-\log \left (e^5+\frac {x}{4}\right )}{40-x+\log \left (e^5+\frac {x}{4}\right )+5 \log (x)}\right ) \] Input:
Integrate[(-1912*x + 81*x^2 - x^3 + E^5*(-7680 + 324*x - 4*x^2) + (-479*x + 10*x^2 + E^5*(-1920 + 40*x))*Log[x] + (-120*E^5 - 30*x)*Log[x]^2 + (-89* x + 2*x^2 + E^5*(-356 + 8*x) + (-44*E^5 - 11*x)*Log[x])*Log[(4*E^5 + x)/4] + (-4*E^5 - x)*Log[(4*E^5 + x)/4]^2)/(1600*x - 80*x^2 + x^3 + E^5*(6400 - 320*x + 4*x^2) + (E^5*(1600 - 40*x) + 400*x - 10*x^2)*Log[x] + (100*E^5 + 25*x)*Log[x]^2 + (E^5*(320 - 8*x) + 80*x - 2*x^2 + (40*E^5 + 10*x)*Log[x] )*Log[(4*E^5 + x)/4] + (4*E^5 + x)*Log[(4*E^5 + x)/4]^2),x]
Output:
-1/5*(x*(6 + (x - Log[E^5 + x/4])/(40 - x + Log[E^5 + x/4] + 5*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.
| \(\displaystyle \int \frac {-x^3+81 x^2+e^5 \left (-4 x^2+324 x-7680\right )+\left (10 x^2-479 x+e^5 (40 x-1920)\right ) \log (x)+\left (2 x^2-89 x+e^5 (8 x-356)+\left (-11 x-44 e^5\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (x+4 e^5\right )\right )-1912 x+\left (-30 x-120 e^5\right ) \log ^2(x)+\left (-x-4 e^5\right ) \log ^2\left (\frac {1}{4} \left (x+4 e^5\right )\right )}{x^3-80 x^2+e^5 \left (4 x^2-320 x+6400\right )+\left (-10 x^2+400 x+e^5 (1600-40 x)\right ) \log (x)+\left (-2 x^2+80 x+e^5 (320-8 x)+\left (10 x+40 e^5\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (x+4 e^5\right )\right )+1600 x+\left (25 x+100 e^5\right ) \log ^2(x)+\left (x+4 e^5\right ) \log ^2\left (\frac {1}{4} \left (x+4 e^5\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-x^3+81 \left (1-\frac {4 e^5}{81}\right ) x^2-1912 \left (1-\frac {81 e^5}{478}\right ) x-\left (x+4 e^5\right ) \log ^2\left (\frac {x}{4}+e^5\right )-30 \left (x+4 e^5\right ) \log ^2(x)+\left (x+4 e^5\right ) \log \left (\frac {x}{4}+e^5\right ) (2 x-11 \log (x)-89)-\left ((479-10 x) x-40 e^5 (x-48)\right ) \log (x)-7680 e^5}{\left (x+4 e^5\right ) \left (-x+\log \left (\frac {x}{4}+e^5\right )+5 \log (x)+40\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (-x^2+2 \left (3-2 e^5\right ) x+20 e^5\right ) \left (x-\log \left (\frac {x}{4}+e^5\right )\right )}{5 \left (x+4 e^5\right ) \left (-x+\log \left (\frac {x}{4}+e^5\right )+5 \log (x)+40\right )^2}+\frac {-2 x^2+\left (1-8 e^5\right ) x+x \log \left (\frac {x}{4}+e^5\right )+4 e^5 \log \left (\frac {x}{4}+e^5\right )}{5 \left (x+4 e^5\right ) \left (-x+\log \left (\frac {x}{4}+e^5\right )+5 \log (x)+40\right )}-\frac {6}{5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{5} \int \frac {x^2}{\left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx-\frac {4}{5} e^5 \int \frac {1}{\left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx+\frac {6}{5} \int \frac {x}{\left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx+\frac {16}{5} e^{10} \int \frac {1}{\left (x+4 e^5\right ) \left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx+\frac {1}{5} \int \frac {x \log \left (\frac {x}{4}+e^5\right )}{\left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx+\frac {4}{5} e^5 \int \frac {\log \left (\frac {x}{4}+e^5\right )}{\left (x+4 e^5\right ) \left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )^2}dx-\frac {1}{5} \left (1-8 e^5\right ) \int \frac {1}{x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40}dx-\frac {8}{5} e^5 \int \frac {1}{x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40}dx+\frac {2}{5} \int \frac {x}{x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40}dx+\frac {4}{5} e^5 \left (1-8 e^5\right ) \int \frac {1}{\left (x+4 e^5\right ) \left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )}dx+\frac {32}{5} e^{10} \int \frac {1}{\left (x+4 e^5\right ) \left (x-\log \left (\frac {x}{4}+e^5\right )-5 \log (x)-40\right )}dx-\frac {6}{5} \int \frac {\log \left (\frac {x}{4}+e^5\right )}{\left (-x+\log \left (\frac {x}{4}+e^5\right )+5 \log (x)+40\right )^2}dx+\frac {1}{5} \int \frac {\log \left (\frac {x}{4}+e^5\right )}{-x+\log \left (\frac {x}{4}+e^5\right )+5 \log (x)+40}dx-\frac {6 x}{5}\) |
Input:
Int[(-1912*x + 81*x^2 - x^3 + E^5*(-7680 + 324*x - 4*x^2) + (-479*x + 10*x ^2 + E^5*(-1920 + 40*x))*Log[x] + (-120*E^5 - 30*x)*Log[x]^2 + (-89*x + 2* x^2 + E^5*(-356 + 8*x) + (-44*E^5 - 11*x)*Log[x])*Log[(4*E^5 + x)/4] + (-4 *E^5 - x)*Log[(4*E^5 + x)/4]^2)/(1600*x - 80*x^2 + x^3 + E^5*(6400 - 320*x + 4*x^2) + (E^5*(1600 - 40*x) + 400*x - 10*x^2)*Log[x] + (100*E^5 + 25*x) *Log[x]^2 + (E^5*(320 - 8*x) + 80*x - 2*x^2 + (40*E^5 + 10*x)*Log[x])*Log[ (4*E^5 + x)/4] + (4*E^5 + x)*Log[(4*E^5 + x)/4]^2),x]
Output:
$Aborted
Time = 15.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88
| method | result | size |
| risch | \(-x +\frac {\left (\ln \left (x \right )+8\right ) x}{x -5 \ln \left (x \right )-\ln \left ({\mathrm e}^{5}+\frac {x}{4}\right )-40}\) | \(29\) |
| default | \(-x +\frac {\left (\ln \left (x \right )+8\right ) x}{x -\ln \left (4 \,{\mathrm e}^{5}+x \right )+2 \ln \left (2\right )-5 \ln \left (x \right )-40}\) | \(33\) |
| parallelrisch | \(-\frac {-48 x +40 \,{\mathrm e}^{5} \ln \left (x \right )-6 x \ln \left (x \right )-8 x \,{\mathrm e}^{5}+320 \,{\mathrm e}^{5}+x^{2}-\ln \left ({\mathrm e}^{5}+\frac {x}{4}\right ) x +8 \,{\mathrm e}^{5} \ln \left ({\mathrm e}^{5}+\frac {x}{4}\right )}{x -5 \ln \left (x \right )-\ln \left ({\mathrm e}^{5}+\frac {x}{4}\right )-40}\) | \(69\) |
Input:
int(((-4*exp(5)-x)*ln(exp(5)+1/4*x)^2+((-44*exp(5)-11*x)*ln(x)+(8*x-356)*e xp(5)+2*x^2-89*x)*ln(exp(5)+1/4*x)+(-120*exp(5)-30*x)*ln(x)^2+((40*x-1920) *exp(5)+10*x^2-479*x)*ln(x)+(-4*x^2+324*x-7680)*exp(5)-x^3+81*x^2-1912*x)/ ((4*exp(5)+x)*ln(exp(5)+1/4*x)^2+((40*exp(5)+10*x)*ln(x)+(-8*x+320)*exp(5) -2*x^2+80*x)*ln(exp(5)+1/4*x)+(100*exp(5)+25*x)*ln(x)^2+((-40*x+1600)*exp( 5)-10*x^2+400*x)*ln(x)+(4*x^2-320*x+6400)*exp(5)+x^3-80*x^2+1600*x),x,meth od=_RETURNVERBOSE)
Output:
-x+(ln(x)+8)*x/(x-5*ln(x)-ln(exp(5)+1/4*x)-40)
Time = 0.08 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=-\frac {x^{2} - 6 \, x \log \left (x\right ) - x \log \left (\frac {1}{4} \, x + e^{5}\right ) - 48 \, x}{x - 5 \, \log \left (x\right ) - \log \left (\frac {1}{4} \, x + e^{5}\right ) - 40} \] Input:
integrate(((-4*exp(5)-x)*log(exp(5)+1/4*x)^2+((-44*exp(5)-11*x)*log(x)+(8* x-356)*exp(5)+2*x^2-89*x)*log(exp(5)+1/4*x)+(-120*exp(5)-30*x)*log(x)^2+(( 40*x-1920)*exp(5)+10*x^2-479*x)*log(x)+(-4*x^2+324*x-7680)*exp(5)-x^3+81*x ^2-1912*x)/((4*exp(5)+x)*log(exp(5)+1/4*x)^2+((40*exp(5)+10*x)*log(x)+(-8* x+320)*exp(5)-2*x^2+80*x)*log(exp(5)+1/4*x)+(100*exp(5)+25*x)*log(x)^2+((- 40*x+1600)*exp(5)-10*x^2+400*x)*log(x)+(4*x^2-320*x+6400)*exp(5)+x^3-80*x^ 2+1600*x),x, algorithm="fricas")
Output:
-(x^2 - 6*x*log(x) - x*log(1/4*x + e^5) - 48*x)/(x - 5*log(x) - log(1/4*x + e^5) - 40)
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=- x + \frac {- x \log {\left (x \right )} - 8 x}{- x + 5 \log {\left (x \right )} + \log {\left (\frac {x}{4} + e^{5} \right )} + 40} \] Input:
integrate(((-4*exp(5)-x)*ln(exp(5)+1/4*x)**2+((-44*exp(5)-11*x)*ln(x)+(8*x -356)*exp(5)+2*x**2-89*x)*ln(exp(5)+1/4*x)+(-120*exp(5)-30*x)*ln(x)**2+((4 0*x-1920)*exp(5)+10*x**2-479*x)*ln(x)+(-4*x**2+324*x-7680)*exp(5)-x**3+81* x**2-1912*x)/((4*exp(5)+x)*ln(exp(5)+1/4*x)**2+((40*exp(5)+10*x)*ln(x)+(-8 *x+320)*exp(5)-2*x**2+80*x)*ln(exp(5)+1/4*x)+(100*exp(5)+25*x)*ln(x)**2+(( -40*x+1600)*exp(5)-10*x**2+400*x)*ln(x)+(4*x**2-320*x+6400)*exp(5)+x**3-80 *x**2+1600*x),x)
Output:
-x + (-x*log(x) - 8*x)/(-x + 5*log(x) + log(x/4 + exp(5)) + 40)
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.52 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=-\frac {x^{2} + 2 \, x {\left (\log \left (2\right ) - 24\right )} - x \log \left (x + 4 \, e^{5}\right ) - 6 \, x \log \left (x\right )}{x + 2 \, \log \left (2\right ) - \log \left (x + 4 \, e^{5}\right ) - 5 \, \log \left (x\right ) - 40} \] Input:
integrate(((-4*exp(5)-x)*log(exp(5)+1/4*x)^2+((-44*exp(5)-11*x)*log(x)+(8* x-356)*exp(5)+2*x^2-89*x)*log(exp(5)+1/4*x)+(-120*exp(5)-30*x)*log(x)^2+(( 40*x-1920)*exp(5)+10*x^2-479*x)*log(x)+(-4*x^2+324*x-7680)*exp(5)-x^3+81*x ^2-1912*x)/((4*exp(5)+x)*log(exp(5)+1/4*x)^2+((40*exp(5)+10*x)*log(x)+(-8* x+320)*exp(5)-2*x^2+80*x)*log(exp(5)+1/4*x)+(100*exp(5)+25*x)*log(x)^2+((- 40*x+1600)*exp(5)-10*x^2+400*x)*log(x)+(4*x^2-320*x+6400)*exp(5)+x^3-80*x^ 2+1600*x),x, algorithm="maxima")
Output:
-(x^2 + 2*x*(log(2) - 24) - x*log(x + 4*e^5) - 6*x*log(x))/(x + 2*log(2) - log(x + 4*e^5) - 5*log(x) - 40)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.67 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=-\frac {5 \, {\left (x + 4 \, e^{5}\right )}^{2} - 16 \, {\left (x + 4 \, e^{5}\right )} e^{5} - 30 \, {\left (x + 4 \, e^{5}\right )} \log \left (x\right ) - 5 \, {\left (x + 4 \, e^{5}\right )} \log \left (\frac {1}{4} \, x + e^{5}\right ) - 4 \, e^{5} \log \left (\frac {1}{4} \, x + e^{5}\right ) - 240 \, x - 16 \, e^{10} - 960 \, e^{5}}{5 \, {\left (x - 5 \, \log \left (x\right ) - \log \left (\frac {1}{4} \, x + e^{5}\right ) - 40\right )}} \] Input:
integrate(((-4*exp(5)-x)*log(exp(5)+1/4*x)^2+((-44*exp(5)-11*x)*log(x)+(8* x-356)*exp(5)+2*x^2-89*x)*log(exp(5)+1/4*x)+(-120*exp(5)-30*x)*log(x)^2+(( 40*x-1920)*exp(5)+10*x^2-479*x)*log(x)+(-4*x^2+324*x-7680)*exp(5)-x^3+81*x ^2-1912*x)/((4*exp(5)+x)*log(exp(5)+1/4*x)^2+((40*exp(5)+10*x)*log(x)+(-8* x+320)*exp(5)-2*x^2+80*x)*log(exp(5)+1/4*x)+(100*exp(5)+25*x)*log(x)^2+((- 40*x+1600)*exp(5)-10*x^2+400*x)*log(x)+(4*x^2-320*x+6400)*exp(5)+x^3-80*x^ 2+1600*x),x, algorithm="giac")
Output:
-1/5*(5*(x + 4*e^5)^2 - 16*(x + 4*e^5)*e^5 - 30*(x + 4*e^5)*log(x) - 5*(x + 4*e^5)*log(1/4*x + e^5) - 4*e^5*log(1/4*x + e^5) - 240*x - 16*e^10 - 960 *e^5)/(x - 5*log(x) - log(1/4*x + e^5) - 40)
Time = 2.86 (sec) , antiderivative size = 187, normalized size of antiderivative = 5.67 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=\ln \left (x\right )-x+\frac {54\,x+180\,{\mathrm {e}}^5}{x^2+\left (4\,{\mathrm {e}}^5-6\right )\,x-20\,{\mathrm {e}}^5}+\frac {\frac {x\,\left (312\,x+1280\,{\mathrm {e}}^5+20\,{\mathrm {e}}^5\,{\ln \left (x\right )}^2+5\,x\,{\ln \left (x\right )}^2-4\,x\,{\mathrm {e}}^5+320\,{\mathrm {e}}^5\,\ln \left (x\right )+79\,x\,\ln \left (x\right )-x^2\right )}{6\,x+20\,{\mathrm {e}}^5-4\,x\,{\mathrm {e}}^5-x^2}+\frac {x\,\ln \left (\frac {x}{4}+{\mathrm {e}}^5\right )\,\left (\ln \left (x\right )+9\right )\,\left (x+4\,{\mathrm {e}}^5\right )}{6\,x+20\,{\mathrm {e}}^5-4\,x\,{\mathrm {e}}^5-x^2}}{\ln \left (\frac {x}{4}+{\mathrm {e}}^5\right )-x+5\,\ln \left (x\right )+40}+\frac {\ln \left (x\right )\,\left (6\,x+20\,{\mathrm {e}}^5\right )}{x^2+\left (4\,{\mathrm {e}}^5-6\right )\,x-20\,{\mathrm {e}}^5} \] Input:
int(-(1912*x + log(x/4 + exp(5))*(89*x + log(x)*(11*x + 44*exp(5)) - 2*x^2 - exp(5)*(8*x - 356)) + exp(5)*(4*x^2 - 324*x + 7680) + log(x/4 + exp(5)) ^2*(x + 4*exp(5)) + log(x)^2*(30*x + 120*exp(5)) - log(x)*(10*x^2 - 479*x + exp(5)*(40*x - 1920)) - 81*x^2 + x^3)/(1600*x + log(x/4 + exp(5))*(80*x + log(x)*(10*x + 40*exp(5)) - 2*x^2 - exp(5)*(8*x - 320)) + exp(5)*(4*x^2 - 320*x + 6400) + log(x/4 + exp(5))^2*(x + 4*exp(5)) + log(x)^2*(25*x + 10 0*exp(5)) - log(x)*(10*x^2 - 400*x + exp(5)*(40*x - 1600)) - 80*x^2 + x^3) ,x)
Output:
log(x) - x + (54*x + 180*exp(5))/(x^2 - 20*exp(5) + x*(4*exp(5) - 6)) + (( x*(312*x + 1280*exp(5) + 20*exp(5)*log(x)^2 + 5*x*log(x)^2 - 4*x*exp(5) + 320*exp(5)*log(x) + 79*x*log(x) - x^2))/(6*x + 20*exp(5) - 4*x*exp(5) - x^ 2) + (x*log(x/4 + exp(5))*(log(x) + 9)*(x + 4*exp(5)))/(6*x + 20*exp(5) - 4*x*exp(5) - x^2))/(log(x/4 + exp(5)) - x + 5*log(x) + 40) + (log(x)*(6*x + 20*exp(5)))/(x^2 - 20*exp(5) + x*(4*exp(5) - 6))
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.58 \[ \int \frac {-1912 x+81 x^2-x^3+e^5 \left (-7680+324 x-4 x^2\right )+\left (-479 x+10 x^2+e^5 (-1920+40 x)\right ) \log (x)+\left (-120 e^5-30 x\right ) \log ^2(x)+\left (-89 x+2 x^2+e^5 (-356+8 x)+\left (-44 e^5-11 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (-4 e^5-x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )}{1600 x-80 x^2+x^3+e^5 \left (6400-320 x+4 x^2\right )+\left (e^5 (1600-40 x)+400 x-10 x^2\right ) \log (x)+\left (100 e^5+25 x\right ) \log ^2(x)+\left (e^5 (320-8 x)+80 x-2 x^2+\left (40 e^5+10 x\right ) \log (x)\right ) \log \left (\frac {1}{4} \left (4 e^5+x\right )\right )+\left (4 e^5+x\right ) \log ^2\left (\frac {1}{4} \left (4 e^5+x\right )\right )} \, dx=\frac {\mathrm {log}\left (4 e^{5}+x \right ) \mathrm {log}\left (e^{5}+\frac {x}{4}\right )+5 \,\mathrm {log}\left (4 e^{5}+x \right ) \mathrm {log}\left (x \right )-\mathrm {log}\left (4 e^{5}+x \right ) x +40 \,\mathrm {log}\left (4 e^{5}+x \right )-\mathrm {log}\left (e^{5}+\frac {x}{4}\right )^{2}-5 \,\mathrm {log}\left (e^{5}+\frac {x}{4}\right ) \mathrm {log}\left (x \right )-88 \,\mathrm {log}\left (e^{5}+\frac {x}{4}\right )-6 \,\mathrm {log}\left (x \right ) x -240 \,\mathrm {log}\left (x \right )+x^{2}-1920}{\mathrm {log}\left (e^{5}+\frac {x}{4}\right )+5 \,\mathrm {log}\left (x \right )-x +40} \] Input:
int(((-4*exp(5)-x)*log(exp(5)+1/4*x)^2+((-44*exp(5)-11*x)*log(x)+(8*x-356) *exp(5)+2*x^2-89*x)*log(exp(5)+1/4*x)+(-120*exp(5)-30*x)*log(x)^2+((40*x-1 920)*exp(5)+10*x^2-479*x)*log(x)+(-4*x^2+324*x-7680)*exp(5)-x^3+81*x^2-191 2*x)/((4*exp(5)+x)*log(exp(5)+1/4*x)^2+((40*exp(5)+10*x)*log(x)+(-8*x+320) *exp(5)-2*x^2+80*x)*log(exp(5)+1/4*x)+(100*exp(5)+25*x)*log(x)^2+((-40*x+1 600)*exp(5)-10*x^2+400*x)*log(x)+(4*x^2-320*x+6400)*exp(5)+x^3-80*x^2+1600 *x),x)
Output:
(log(4*e**5 + x)*log((4*e**5 + x)/4) + 5*log(4*e**5 + x)*log(x) - log(4*e* *5 + x)*x + 40*log(4*e**5 + x) - log((4*e**5 + x)/4)**2 - 5*log((4*e**5 + x)/4)*log(x) - 88*log((4*e**5 + x)/4) - 6*log(x)*x - 240*log(x) + x**2 - 1 920)/(log((4*e**5 + x)/4) + 5*log(x) - x + 40)