Integrand size = 192, antiderivative size = 23 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x+\log \left (5+x+\log \left (-e^4+x+\frac {x}{4+5 x}\right )\right ) \] Output:
x+ln(5+ln(x+x/(4+5*x)-exp(4))+x)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x+\log \left (5+x+\log \left (-e^4+\frac {5 x (1+x)}{4+5 x}\right )\right ) \] Input:
Integrate[(-20 - 160*x - 315*x^2 - 195*x^3 - 25*x^4 + E^4*(96 + 256*x + 19 0*x^2 + 25*x^3) + (-20*x - 45*x^2 - 25*x^3 + E^4*(16 + 40*x + 25*x^2))*Log [(E^4*(-4 - 5*x) + 5*x + 5*x^2)/(4 + 5*x)])/(-100*x - 245*x^2 - 170*x^3 - 25*x^4 + E^4*(80 + 216*x + 165*x^2 + 25*x^3) + (-20*x - 45*x^2 - 25*x^3 + E^4*(16 + 40*x + 25*x^2))*Log[(E^4*(-4 - 5*x) + 5*x + 5*x^2)/(4 + 5*x)]),x ]
Output:
x + Log[5 + x + Log[-E^4 + (5*x*(1 + x))/(4 + 5*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.
| \(\displaystyle \int \frac {-25 x^4-195 x^3-315 x^2+e^4 \left (25 x^3+190 x^2+256 x+96\right )+\left (-25 x^3-45 x^2+e^4 \left (25 x^2+40 x+16\right )-20 x\right ) \log \left (\frac {5 x^2+5 x+e^4 (-5 x-4)}{5 x+4}\right )-160 x-20}{-25 x^4-170 x^3-245 x^2+e^4 \left (25 x^3+165 x^2+216 x+80\right )+\left (-25 x^3-45 x^2+e^4 \left (25 x^2+40 x+16\right )-20 x\right ) \log \left (\frac {5 x^2+5 x+e^4 (-5 x-4)}{5 x+4}\right )-100 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-25 x^4-195 x^3-315 x^2+e^4 \left (25 x^3+190 x^2+256 x+96\right )+\left (-25 x^3-45 x^2+e^4 \left (25 x^2+40 x+16\right )-20 x\right ) \log \left (\frac {5 x^2+5 x+e^4 (-5 x-4)}{5 x+4}\right )-160 x-20}{(5 x+4) \left (-5 x^2-5 \left (1-e^4\right ) x+4 e^4\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {5 \left (-25 x^4-195 \left (1-\frac {5 e^4}{39}\right ) x^3-25 x^3 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-315 \left (1-\frac {38 e^4}{63}\right ) x^2-45 \left (1-\frac {5 e^4}{9}\right ) x^2 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-160 \left (1-\frac {8 e^4}{5}\right ) x-20 \left (1-2 e^4\right ) x \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+16 e^4 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-20 \left (1-\frac {24 e^4}{5}\right )\right )}{4 (5 x+4) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}+\frac {\left (5 x-5 e^4+1\right ) \left (-25 x^4-195 \left (1-\frac {5 e^4}{39}\right ) x^3-25 x^3 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-315 \left (1-\frac {38 e^4}{63}\right ) x^2-45 \left (1-\frac {5 e^4}{9}\right ) x^2 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-160 \left (1-\frac {8 e^4}{5}\right ) x-20 \left (1-2 e^4\right ) x \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+16 e^4 \log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )-20 \left (1-\frac {24 e^4}{5}\right )\right )}{4 \left (-5 x^2-5 \left (1-e^4\right ) x+4 e^4\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \left (1-e^4\right ) \int \frac {1}{x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5}dx+\left (6-5 e^4\right ) \int \frac {1}{x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5}dx-10 \left (1-e^4\right ) \sqrt {\frac {5}{5+6 e^4+5 e^8}} \int \frac {1}{\left (-10 x+\sqrt {5 \left (5+6 e^4+5 e^8\right )}+5 e^4-5\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx-5 \int \frac {1}{(5 x+4) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx+10 \left (1-\left (1-e^4\right ) \sqrt {\frac {5}{5+6 e^4+5 e^8}}\right ) \int \frac {1}{\left (10 x-\sqrt {5 \left (5+6 e^4+5 e^8\right )}+5 \left (1-e^4\right )\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx-10 \left (1-e^4\right ) \sqrt {\frac {5}{5+6 e^4+5 e^8}} \int \frac {1}{\left (10 x+\sqrt {5 \left (5+6 e^4+5 e^8\right )}-5 e^4+5\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx+10 \left (1+\left (1-e^4\right ) \sqrt {\frac {5}{5+6 e^4+5 e^8}}\right ) \int \frac {1}{\left (10 x+\sqrt {5 \left (5+6 e^4+5 e^8\right )}+5 \left (1-e^4\right )\right ) \left (x+\log \left (\frac {5 x (x+1)}{5 x+4}-e^4\right )+5\right )}dx+\left (1-5 e^4\right ) x+5 e^4 x\) |
Input:
Int[(-20 - 160*x - 315*x^2 - 195*x^3 - 25*x^4 + E^4*(96 + 256*x + 190*x^2 + 25*x^3) + (-20*x - 45*x^2 - 25*x^3 + E^4*(16 + 40*x + 25*x^2))*Log[(E^4* (-4 - 5*x) + 5*x + 5*x^2)/(4 + 5*x)])/(-100*x - 245*x^2 - 170*x^3 - 25*x^4 + E^4*(80 + 216*x + 165*x^2 + 25*x^3) + (-20*x - 45*x^2 - 25*x^3 + E^4*(1 6 + 40*x + 25*x^2))*Log[(E^4*(-4 - 5*x) + 5*x + 5*x^2)/(4 + 5*x)]),x]
Output:
$Aborted
Time = 4.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43
| method | result | size |
| norman | \(x +\ln \left (x +\ln \left (\frac {\left (-5 x -4\right ) {\mathrm e}^{4}+5 x^{2}+5 x}{4+5 x}\right )+5\right )\) | \(33\) |
| risch | \(x +\ln \left (x +\ln \left (\frac {\left (-5 x -4\right ) {\mathrm e}^{4}+5 x^{2}+5 x}{4+5 x}\right )+5\right )\) | \(33\) |
| parallelrisch | \(-\frac {\left (-256 \ln \left (x +\ln \left (\frac {\left (-5 x -4\right ) {\mathrm e}^{4}+5 x^{2}+5 x}{4+5 x}\right )+5\right ) {\mathrm e}^{8}-256 x \,{\mathrm e}^{8}\right ) {\mathrm e}^{-8}}{256}\) | \(51\) |
Input:
int((((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*ln(((-5*x-4)*exp(4)+5*x^ 2+5*x)/(4+5*x))+(25*x^3+190*x^2+256*x+96)*exp(4)-25*x^4-195*x^3-315*x^2-16 0*x-20)/(((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*ln(((-5*x-4)*exp(4)+ 5*x^2+5*x)/(4+5*x))+(25*x^3+165*x^2+216*x+80)*exp(4)-25*x^4-170*x^3-245*x^ 2-100*x),x,method=_RETURNVERBOSE)
Output:
x+ln(x+ln(((-5*x-4)*exp(4)+5*x^2+5*x)/(4+5*x))+5)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x + \log \left (x + \log \left (\frac {5 \, x^{2} - {\left (5 \, x + 4\right )} e^{4} + 5 \, x}{5 \, x + 4}\right ) + 5\right ) \] Input:
integrate((((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4)*exp( 4)+5*x^2+5*x)/(4+5*x))+(25*x^3+190*x^2+256*x+96)*exp(4)-25*x^4-195*x^3-315 *x^2-160*x-20)/(((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4) *exp(4)+5*x^2+5*x)/(4+5*x))+(25*x^3+165*x^2+216*x+80)*exp(4)-25*x^4-170*x^ 3-245*x^2-100*x),x, algorithm="fricas")
Output:
x + log(x + log((5*x^2 - (5*x + 4)*e^4 + 5*x)/(5*x + 4)) + 5)
Time = 0.37 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x + \log {\left (x + \log {\left (\frac {5 x^{2} + 5 x + \left (- 5 x - 4\right ) e^{4}}{5 x + 4} \right )} + 5 \right )} \] Input:
integrate((((25*x**2+40*x+16)*exp(4)-25*x**3-45*x**2-20*x)*ln(((-5*x-4)*ex p(4)+5*x**2+5*x)/(4+5*x))+(25*x**3+190*x**2+256*x+96)*exp(4)-25*x**4-195*x **3-315*x**2-160*x-20)/(((25*x**2+40*x+16)*exp(4)-25*x**3-45*x**2-20*x)*ln (((-5*x-4)*exp(4)+5*x**2+5*x)/(4+5*x))+(25*x**3+165*x**2+216*x+80)*exp(4)- 25*x**4-170*x**3-245*x**2-100*x),x)
Output:
x + log(x + log((5*x**2 + 5*x + (-5*x - 4)*exp(4))/(5*x + 4)) + 5)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x + \log \left (x + \log \left (5 \, x^{2} - 5 \, x {\left (e^{4} - 1\right )} - 4 \, e^{4}\right ) - \log \left (5 \, x + 4\right ) + 5\right ) \] Input:
integrate((((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4)*exp( 4)+5*x^2+5*x)/(4+5*x))+(25*x^3+190*x^2+256*x+96)*exp(4)-25*x^4-195*x^3-315 *x^2-160*x-20)/(((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4) *exp(4)+5*x^2+5*x)/(4+5*x))+(25*x^3+165*x^2+216*x+80)*exp(4)-25*x^4-170*x^ 3-245*x^2-100*x),x, algorithm="maxima")
Output:
x + log(x + log(5*x^2 - 5*x*(e^4 - 1) - 4*e^4) - log(5*x + 4) + 5)
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x + \log \left (x + \log \left (\frac {5 \, x^{2} - 5 \, x e^{4} + 5 \, x - 4 \, e^{4}}{5 \, x + 4}\right ) + 5\right ) \] Input:
integrate((((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4)*exp( 4)+5*x^2+5*x)/(4+5*x))+(25*x^3+190*x^2+256*x+96)*exp(4)-25*x^4-195*x^3-315 *x^2-160*x-20)/(((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4) *exp(4)+5*x^2+5*x)/(4+5*x))+(25*x^3+165*x^2+216*x+80)*exp(4)-25*x^4-170*x^ 3-245*x^2-100*x),x, algorithm="giac")
Output:
x + log(x + log((5*x^2 - 5*x*e^4 + 5*x - 4*e^4)/(5*x + 4)) + 5)
Time = 122.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=x+\ln \left (x+\ln \left (\frac {5\,x+5\,x^2-{\mathrm {e}}^4\,\left (5\,x+4\right )}{5\,x+4}\right )+5\right ) \] Input:
int((160*x + log((5*x + 5*x^2 - exp(4)*(5*x + 4))/(5*x + 4))*(20*x - exp(4 )*(40*x + 25*x^2 + 16) + 45*x^2 + 25*x^3) - exp(4)*(256*x + 190*x^2 + 25*x ^3 + 96) + 315*x^2 + 195*x^3 + 25*x^4 + 20)/(100*x + log((5*x + 5*x^2 - ex p(4)*(5*x + 4))/(5*x + 4))*(20*x - exp(4)*(40*x + 25*x^2 + 16) + 45*x^2 + 25*x^3) - exp(4)*(216*x + 165*x^2 + 25*x^3 + 80) + 245*x^2 + 170*x^3 + 25* x^4),x)
Output:
x + log(x + log((5*x + 5*x^2 - exp(4)*(5*x + 4))/(5*x + 4)) + 5)
Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int \frac {-20-160 x-315 x^2-195 x^3-25 x^4+e^4 \left (96+256 x+190 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )}{-100 x-245 x^2-170 x^3-25 x^4+e^4 \left (80+216 x+165 x^2+25 x^3\right )+\left (-20 x-45 x^2-25 x^3+e^4 \left (16+40 x+25 x^2\right )\right ) \log \left (\frac {e^4 (-4-5 x)+5 x+5 x^2}{4+5 x}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {-5 e^{4} x -4 e^{4}+5 x^{2}+5 x}{5 x +4}\right )+x +5\right )+x \] Input:
int((((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4)*exp(4)+5*x ^2+5*x)/(4+5*x))+(25*x^3+190*x^2+256*x+96)*exp(4)-25*x^4-195*x^3-315*x^2-1 60*x-20)/(((25*x^2+40*x+16)*exp(4)-25*x^3-45*x^2-20*x)*log(((-5*x-4)*exp(4 )+5*x^2+5*x)/(4+5*x))+(25*x^3+165*x^2+216*x+80)*exp(4)-25*x^4-170*x^3-245* x^2-100*x),x)
Output:
log(log(( - 5*e**4*x - 4*e**4 + 5*x**2 + 5*x)/(5*x + 4)) + x + 5) + x