Integrand size = 109, antiderivative size = 27 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=\frac {8 x}{1-\log \left (5+x-\frac {-\frac {20}{x}+2 x}{x}\right )} \] Output:
8*x/(1-ln(x-(2*x-20/x)/x+5))
Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 x}{-1+\log \left (3+\frac {20}{x^2}+x\right )} \] Input:
Integrate[(-160 + 24*x^2 + 16*x^3 + (-160 - 24*x^2 - 8*x^3)*Log[(20 + 3*x^ 2 + x^3)/x^2])/(20 + 3*x^2 + x^3 + (-40 - 6*x^2 - 2*x^3)*Log[(20 + 3*x^2 + x^3)/x^2] + (20 + 3*x^2 + x^3)*Log[(20 + 3*x^2 + x^3)/x^2]^2),x]
Output:
(-8*x)/(-1 + Log[3 + 20/x^2 + 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.
\(\displaystyle \int \frac {16 x^3+24 x^2+\left (-8 x^3-24 x^2-160\right ) \log \left (\frac {x^3+3 x^2+20}{x^2}\right )-160}{x^3+3 x^2+\left (x^3+3 x^2+20\right ) \log ^2\left (\frac {x^3+3 x^2+20}{x^2}\right )+\left (-2 x^3-6 x^2-40\right ) \log \left (\frac {x^3+3 x^2+20}{x^2}\right )+20} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 \left (2 x^3+3 x^2-\left (x^3+3 x^2+20\right ) \log \left (\frac {20}{x^2}+x+3\right )-20\right )}{\left (x^3+3 x^2+20\right ) \left (1-\log \left (\frac {20}{x^2}+x+3\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int -\frac {-2 x^3-3 x^2+\left (x^3+3 x^2+20\right ) \log \left (x+3+\frac {20}{x^2}\right )+20}{\left (x^3+3 x^2+20\right ) \left (1-\log \left (x+3+\frac {20}{x^2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -8 \int \frac {-2 x^3-3 x^2+\left (x^3+3 x^2+20\right ) \log \left (x+3+\frac {20}{x^2}\right )+20}{\left (x^3+3 x^2+20\right ) \left (1-\log \left (x+3+\frac {20}{x^2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -8 \int \left (\frac {40-x^3}{\left (x^3+3 x^2+20\right ) \left (\log \left (x+3+\frac {20}{x^2}\right )-1\right )^2}+\frac {1}{\log \left (x+3+\frac {20}{x^2}\right )-1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 \left (-\int \frac {1}{\left (\log \left (x+3+\frac {20}{x^2}\right )-1\right )^2}dx+\int \frac {1}{\log \left (x+3+\frac {20}{x^2}\right )-1}dx+60 \int \frac {1}{\left (x^3+3 x^2+20\right ) \left (\log \left (x+3+\frac {20}{x^2}\right )-1\right )^2}dx+3 \int \frac {x^2}{\left (x^3+3 x^2+20\right ) \left (\log \left (x+3+\frac {20}{x^2}\right )-1\right )^2}dx\right )\) |
Input:
Int[(-160 + 24*x^2 + 16*x^3 + (-160 - 24*x^2 - 8*x^3)*Log[(20 + 3*x^2 + x^ 3)/x^2])/(20 + 3*x^2 + x^3 + (-40 - 6*x^2 - 2*x^3)*Log[(20 + 3*x^2 + x^3)/ x^2] + (20 + 3*x^2 + x^3)*Log[(20 + 3*x^2 + x^3)/x^2]^2),x]
Output:
$Aborted
Time = 0.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
norman | \(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\) | \(23\) |
risch | \(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\) | \(23\) |
parallelrisch | \(-\frac {8 x}{\ln \left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1}\) | \(23\) |
Input:
int(((-8*x^3-24*x^2-160)*ln((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+3 *x^2+20)*ln((x^3+3*x^2+20)/x^2)^2+(-2*x^3-6*x^2-40)*ln((x^3+3*x^2+20)/x^2) +x^3+3*x^2+20),x,method=_RETURNVERBOSE)
Output:
-8*x/(ln((x^3+3*x^2+20)/x^2)-1)
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (\frac {x^{3} + 3 \, x^{2} + 20}{x^{2}}\right ) - 1} \] Input:
integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/ ((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2 +20)/x^2)+x^3+3*x^2+20),x, algorithm="fricas")
Output:
-8*x/(log((x^3 + 3*x^2 + 20)/x^2) - 1)
Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=- \frac {8 x}{\log {\left (\frac {x^{3} + 3 x^{2} + 20}{x^{2}} \right )} - 1} \] Input:
integrate(((-8*x**3-24*x**2-160)*ln((x**3+3*x**2+20)/x**2)+16*x**3+24*x**2 -160)/((x**3+3*x**2+20)*ln((x**3+3*x**2+20)/x**2)**2+(-2*x**3-6*x**2-40)*l n((x**3+3*x**2+20)/x**2)+x**3+3*x**2+20),x)
Output:
-8*x/(log((x**3 + 3*x**2 + 20)/x**2) - 1)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (x^{3} + 3 \, x^{2} + 20\right ) - 2 \, \log \left (x\right ) - 1} \] Input:
integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/ ((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2 +20)/x^2)+x^3+3*x^2+20),x, algorithm="maxima")
Output:
-8*x/(log(x^3 + 3*x^2 + 20) - 2*log(x) - 1)
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 \, x}{\log \left (\frac {x^{3} + 3 \, x^{2} + 20}{x^{2}}\right ) - 1} \] Input:
integrate(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/ ((x^3+3*x^2+20)*log((x^3+3*x^2+20)/x^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2 +20)/x^2)+x^3+3*x^2+20),x, algorithm="giac")
Output:
-8*x/(log((x^3 + 3*x^2 + 20)/x^2) - 1)
Time = 3.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8\,\left (x-3\,\ln \left (\frac {x^3+3\,x^2+20}{x^2}\right )+3\right )}{\ln \left (\frac {x^3+3\,x^2+20}{x^2}\right )-1} \] Input:
int(-(log((3*x^2 + x^3 + 20)/x^2)*(24*x^2 + 8*x^3 + 160) - 24*x^2 - 16*x^3 + 160)/(log((3*x^2 + x^3 + 20)/x^2)^2*(3*x^2 + x^3 + 20) - log((3*x^2 + x ^3 + 20)/x^2)*(6*x^2 + 2*x^3 + 40) + 3*x^2 + x^3 + 20),x)
Output:
-(8*(x - 3*log((3*x^2 + x^3 + 20)/x^2) + 3))/(log((3*x^2 + x^3 + 20)/x^2) - 1)
Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-160+24 x^2+16 x^3+\left (-160-24 x^2-8 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )}{20+3 x^2+x^3+\left (-40-6 x^2-2 x^3\right ) \log \left (\frac {20+3 x^2+x^3}{x^2}\right )+\left (20+3 x^2+x^3\right ) \log ^2\left (\frac {20+3 x^2+x^3}{x^2}\right )} \, dx=-\frac {8 x}{\mathrm {log}\left (\frac {x^{3}+3 x^{2}+20}{x^{2}}\right )-1} \] Input:
int(((-8*x^3-24*x^2-160)*log((x^3+3*x^2+20)/x^2)+16*x^3+24*x^2-160)/((x^3+ 3*x^2+20)*log((x^3+3*x^2+20)/x^2)^2+(-2*x^3-6*x^2-40)*log((x^3+3*x^2+20)/x ^2)+x^3+3*x^2+20),x)
Output:
( - 8*x)/(log((x**3 + 3*x**2 + 20)/x**2) - 1)