Integrand size = 120, antiderivative size = 22 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \left (2+x+x^2+x (\log (3)+\log (25))+\log (x)\right )} \] Output:
x/(10+5*x+5*x*(ln(3)+2*ln(5))+5*x^2+5*ln(x))
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \left (2+x+x^2+x \log (75)+\log (x)\right )} \] Input:
Integrate[(1 - x^2 + Log[x])/(20 + 20*x + 25*x^2 + 10*x^3 + 5*x^4 + (20*x + 10*x^2 + 10*x^3)*Log[3] + 5*x^2*Log[3]^2 + (20*x + 10*x^2 + 10*x^3 + 10* x^2*Log[3])*Log[25] + 5*x^2*Log[25]^2 + (20 + 10*x + 10*x^2 + 10*x*Log[3] + 10*x*Log[25])*Log[x] + 5*Log[x]^2),x]
Output:
x/(5*(2 + x + x^2 + x*Log[75] + 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.
\(\displaystyle \int \frac {-x^2+\log (x)+1}{5 x^4+10 x^3+25 x^2+5 x^2 \log ^2(25)+5 x^2 \log ^2(3)+\left (10 x^2+10 x+10 x \log (25)+10 x \log (3)+20\right ) \log (x)+\log (25) \left (10 x^3+10 x^2+10 x^2 \log (3)+20 x\right )+\left (10 x^3+10 x^2+20 x\right ) \log (3)+20 x+5 \log ^2(x)+20} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-x^2+\log (x)+1}{5 x^4+10 x^3+5 x^2 \log ^2(25)+x^2 \left (25+5 \log ^2(3)\right )+\left (10 x^2+10 x+10 x \log (25)+10 x \log (3)+20\right ) \log (x)+\log (25) \left (10 x^3+10 x^2+10 x^2 \log (3)+20 x\right )+\left (10 x^3+10 x^2+20 x\right ) \log (3)+20 x+5 \log ^2(x)+20}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-x^2+\log (x)+1}{5 x^4+10 x^3+x^2 \left (25+5 \log ^2(3)+5 \log ^2(25)\right )+\left (10 x^2+10 x+10 x \log (25)+10 x \log (3)+20\right ) \log (x)+\log (25) \left (10 x^3+10 x^2+10 x^2 \log (3)+20 x\right )+\left (10 x^3+10 x^2+20 x\right ) \log (3)+20 x+5 \log ^2(x)+20}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-x^2+\log (x)+1}{5 \left (x^2+x (1+\log (75))+\log (x)+2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {-x^2+\log (x)+1}{\left (x^2+(1+\log (75)) x+\log (x)+2\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {-2 x^2-(1+\log (75)) x-1}{\left (x^2+(1+\log (75)) x+\log (x)+2\right )^2}+\frac {1}{x^2+(1+\log (75)) x+\log (x)+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{5} \left (-\int \frac {1}{\left (x^2+(1+\log (75)) x+\log (x)+2\right )^2}dx-(1+\log (75)) \int \frac {x}{\left (x^2+(1+\log (75)) x+\log (x)+2\right )^2}dx-2 \int \frac {x^2}{\left (x^2+(1+\log (75)) x+\log (x)+2\right )^2}dx+\int \frac {1}{x^2+(1+\log (75)) x+\log (x)+2}dx\right )\) |
Input:
Int[(1 - x^2 + Log[x])/(20 + 20*x + 25*x^2 + 10*x^3 + 5*x^4 + (20*x + 10*x ^2 + 10*x^3)*Log[3] + 5*x^2*Log[3]^2 + (20*x + 10*x^2 + 10*x^3 + 10*x^2*Lo g[3])*Log[25] + 5*x^2*Log[25]^2 + (20 + 10*x + 10*x^2 + 10*x*Log[3] + 10*x *Log[25])*Log[x] + 5*Log[x]^2),x]
Output:
$Aborted
Time = 0.84 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {x}{10 x \ln \left (5\right )+5 x \ln \left (3\right )+5 x^{2}+5 \ln \left (x \right )+5 x +10}\) | \(23\) |
norman | \(\frac {x}{10 x \ln \left (5\right )+5 x \ln \left (3\right )+5 x^{2}+5 \ln \left (x \right )+5 x +10}\) | \(23\) |
risch | \(\frac {x}{10 x \ln \left (5\right )+5 x \ln \left (3\right )+5 x^{2}+5 \ln \left (x \right )+5 x +10}\) | \(23\) |
parallelrisch | \(\frac {x}{10 x \ln \left (5\right )+5 x \ln \left (3\right )+5 x^{2}+5 \ln \left (x \right )+5 x +10}\) | \(23\) |
Input:
int((ln(x)-x^2+1)/(5*ln(x)^2+(20*x*ln(5)+10*x*ln(3)+10*x^2+10*x+20)*ln(x)+ 20*x^2*ln(5)^2+2*(10*x^2*ln(3)+10*x^3+10*x^2+20*x)*ln(5)+5*x^2*ln(3)^2+(10 *x^3+10*x^2+20*x)*ln(3)+5*x^4+10*x^3+25*x^2+20*x+20),x,method=_RETURNVERBO SE)
Output:
1/5*x/(2*x*ln(5)+x*ln(3)+x^2+ln(x)+x+2)
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + x \log \left (3\right ) + x + \log \left (x\right ) + 2\right )}} \] Input:
integrate((log(x)-x^2+1)/(5*log(x)^2+(20*x*log(5)+10*x*log(3)+10*x^2+10*x+ 20)*log(x)+20*x^2*log(5)^2+2*(10*x^2*log(3)+10*x^3+10*x^2+20*x)*log(5)+5*x ^2*log(3)^2+(10*x^3+10*x^2+20*x)*log(3)+5*x^4+10*x^3+25*x^2+20*x+20),x, al gorithm="fricas")
Output:
1/5*x/(x^2 + 2*x*log(5) + x*log(3) + x + log(x) + 2)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 x^{2} + 5 x + 5 x \log {\left (3 \right )} + 10 x \log {\left (5 \right )} + 5 \log {\left (x \right )} + 10} \] Input:
integrate((ln(x)-x**2+1)/(5*ln(x)**2+(20*x*ln(5)+10*x*ln(3)+10*x**2+10*x+2 0)*ln(x)+20*x**2*ln(5)**2+2*(10*x**2*ln(3)+10*x**3+10*x**2+20*x)*ln(5)+5*x **2*ln(3)**2+(10*x**3+10*x**2+20*x)*ln(3)+5*x**4+10*x**3+25*x**2+20*x+20), x)
Output:
x/(5*x**2 + 5*x + 5*x*log(3) + 10*x*log(5) + 5*log(x) + 10)
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \, {\left (x^{2} + x {\left (2 \, \log \left (5\right ) + \log \left (3\right ) + 1\right )} + \log \left (x\right ) + 2\right )}} \] Input:
integrate((log(x)-x^2+1)/(5*log(x)^2+(20*x*log(5)+10*x*log(3)+10*x^2+10*x+ 20)*log(x)+20*x^2*log(5)^2+2*(10*x^2*log(3)+10*x^3+10*x^2+20*x)*log(5)+5*x ^2*log(3)^2+(10*x^3+10*x^2+20*x)*log(3)+5*x^4+10*x^3+25*x^2+20*x+20),x, al gorithm="maxima")
Output:
1/5*x/(x^2 + x*(2*log(5) + log(3) + 1) + log(x) + 2)
Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + x \log \left (3\right ) + x + \log \left (x\right ) + 2\right )}} \] Input:
integrate((log(x)-x^2+1)/(5*log(x)^2+(20*x*log(5)+10*x*log(3)+10*x^2+10*x+ 20)*log(x)+20*x^2*log(5)^2+2*(10*x^2*log(3)+10*x^3+10*x^2+20*x)*log(5)+5*x ^2*log(3)^2+(10*x^3+10*x^2+20*x)*log(3)+5*x^4+10*x^3+25*x^2+20*x+20),x, al gorithm="giac")
Output:
1/5*x/(x^2 + 2*x*log(5) + x*log(3) + x + log(x) + 2)
Time = 2.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5\,\left (\ln \left (x\right )+x\,\left (\ln \left (75\right )+1\right )+x^2+2\right )} \] Input:
int((log(x) - x^2 + 1)/(20*x + 5*x^2*log(3)^2 + 20*x^2*log(5)^2 + 2*log(5) *(20*x + 10*x^2*log(3) + 10*x^2 + 10*x^3) + log(x)*(10*x + 10*x*log(3) + 2 0*x*log(5) + 10*x^2 + 20) + 5*log(x)^2 + log(3)*(20*x + 10*x^2 + 10*x^3) + 25*x^2 + 10*x^3 + 5*x^4 + 20),x)
Output:
x/(5*(log(x) + x*(log(75) + 1) + x^2 + 2))
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1-x^2+\log (x)}{20+20 x+25 x^2+10 x^3+5 x^4+\left (20 x+10 x^2+10 x^3\right ) \log (3)+5 x^2 \log ^2(3)+\left (20 x+10 x^2+10 x^3+10 x^2 \log (3)\right ) \log (25)+5 x^2 \log ^2(25)+\left (20+10 x+10 x^2+10 x \log (3)+10 x \log (25)\right ) \log (x)+5 \log ^2(x)} \, dx=\frac {x}{5 \,\mathrm {log}\left (x \right )+10 \,\mathrm {log}\left (5\right ) x +5 \,\mathrm {log}\left (3\right ) x +5 x^{2}+5 x +10} \] Input:
int((log(x)-x^2+1)/(5*log(x)^2+(20*x*log(5)+10*x*log(3)+10*x^2+10*x+20)*lo g(x)+20*x^2*log(5)^2+2*(10*x^2*log(3)+10*x^3+10*x^2+20*x)*log(5)+5*x^2*log (3)^2+(10*x^3+10*x^2+20*x)*log(3)+5*x^4+10*x^3+25*x^2+20*x+20),x)
Output:
x/(5*(log(x) + 2*log(5)*x + log(3)*x + x**2 + x + 2))