Integrand size = 107, antiderivative size = 21 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log ^2(5)}{x^2 (1+x+\log ((-3-x) x))} \] Output:
1/x^2/(1+x+ln(x*(-3-x)))*ln(5)^2
Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log ^2(5)}{x^2 (1+x+\log (-x (3+x)))} \] Input:
Integrate[((-9 - 13*x - 3*x^2)*Log[5]^2 + (-6 - 2*x)*Log[5]^2*Log[-3*x - x ^2])/(3*x^3 + 7*x^4 + 5*x^5 + x^6 + (6*x^3 + 8*x^4 + 2*x^5)*Log[-3*x - x^2 ] + (3*x^3 + x^4)*Log[-3*x - x^2]^2),x]
Output:
Log[5]^2/(x^2*(1 + x + Log[-(x*(3 + 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.
\(\displaystyle \int \frac {\left (-3 x^2-13 x-9\right ) \log ^2(5)+(-2 x-6) \log ^2(5) \log \left (-x^2-3 x\right )}{x^6+5 x^5+7 x^4+3 x^3+\left (x^4+3 x^3\right ) \log ^2\left (-x^2-3 x\right )+\left (2 x^5+8 x^4+6 x^3\right ) \log \left (-x^2-3 x\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\log ^2(5) \left (-3 x^2-13 x-2 (x+3) \log (-x (x+3))-9\right )}{x^3 (x+3) (x+\log (-x (x+3))+1)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \log ^2(5) \int -\frac {3 x^2+13 x+2 (x+3) \log (-x (x+3))+9}{x^3 (x+3) (x+\log (-x (x+3))+1)^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\log ^2(5) \int \frac {3 x^2+13 x+2 (x+3) \log (-x (x+3))+9}{x^3 (x+3) (x+\log (-x (x+3))+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\log ^2(5) \int \left (\frac {x^2+5 x+3}{x^3 (x+3) (x+\log (-x (x+3))+1)^2}+\frac {2}{x^3 (x+\log (-x (x+3))+1)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\log ^2(5) \left (\int \frac {1}{x^3 (x+\log (-x (x+3))+1)^2}dx+2 \int \frac {1}{x^3 (x+\log (-x (x+3))+1)}dx+\frac {4}{3} \int \frac {1}{x^2 (x+\log (-x (x+3))+1)^2}dx-\frac {1}{9} \int \frac {1}{x (x+\log (-x (x+3))+1)^2}dx+\frac {1}{9} \int \frac {1}{(x+3) (x+\log (-x (x+3))+1)^2}dx\right )\) |
Input:
Int[((-9 - 13*x - 3*x^2)*Log[5]^2 + (-6 - 2*x)*Log[5]^2*Log[-3*x - x^2])/( 3*x^3 + 7*x^4 + 5*x^5 + x^6 + (6*x^3 + 8*x^4 + 2*x^5)*Log[-3*x - x^2] + (3 *x^3 + x^4)*Log[-3*x - x^2]^2),x]
Output:
$Aborted
Time = 0.47 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
norman | \(\frac {\ln \left (5\right )^{2}}{x^{2} \left (\ln \left (-x^{2}-3 x \right )+x +1\right )}\) | \(24\) |
risch | \(\frac {\ln \left (5\right )^{2}}{x^{2} \left (\ln \left (-x^{2}-3 x \right )+x +1\right )}\) | \(24\) |
parallelrisch | \(\frac {\ln \left (5\right )^{2}}{x^{2} \left (\ln \left (-x^{2}-3 x \right )+x +1\right )}\) | \(24\) |
Input:
int(((-2*x-6)*ln(5)^2*ln(-x^2-3*x)+(-3*x^2-13*x-9)*ln(5)^2)/((x^4+3*x^3)*l n(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*ln(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3),x,me thod=_RETURNVERBOSE)
Output:
ln(5)^2/x^2/(ln(-x^2-3*x)+x+1)
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log \left (5\right )^{2}}{x^{3} + x^{2} \log \left (-x^{2} - 3 \, x\right ) + x^{2}} \] Input:
integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4 +3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+ 3*x^3),x, algorithm="fricas")
Output:
log(5)^2/(x^3 + x^2*log(-x^2 - 3*x) + x^2)
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log {\left (5 \right )}^{2}}{x^{3} + x^{2} \log {\left (- x^{2} - 3 x \right )} + x^{2}} \] Input:
integrate(((-2*x-6)*ln(5)**2*ln(-x**2-3*x)+(-3*x**2-13*x-9)*ln(5)**2)/((x* *4+3*x**3)*ln(-x**2-3*x)**2+(2*x**5+8*x**4+6*x**3)*ln(-x**2-3*x)+x**6+5*x* *5+7*x**4+3*x**3),x)
Output:
log(5)**2/(x**3 + x**2*log(-x**2 - 3*x) + x**2)
Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log \left (5\right )^{2}}{x^{3} + x^{2} \log \left (x\right ) + x^{2} \log \left (-x - 3\right ) + x^{2}} \] Input:
integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4 +3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+ 3*x^3),x, algorithm="maxima")
Output:
log(5)^2/(x^3 + x^2*log(x) + x^2*log(-x - 3) + x^2)
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\log \left (5\right )^{2}}{x^{3} + x^{2} \log \left (-x^{2} - 3 \, x\right ) + x^{2}} \] Input:
integrate(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4 +3*x^3)*log(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+ 3*x^3),x, algorithm="giac")
Output:
log(5)^2/(x^3 + x^2*log(-x^2 - 3*x) + x^2)
Time = 3.66 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {{\ln \left (5\right )}^2}{x^2\,\left (x+\ln \left (-x^2-3\,x\right )+1\right )} \] Input:
int(-(log(5)^2*(13*x + 3*x^2 + 9) + log(5)^2*log(- 3*x - x^2)*(2*x + 6))/( log(- 3*x - x^2)^2*(3*x^3 + x^4) + log(- 3*x - x^2)*(6*x^3 + 8*x^4 + 2*x^5 ) + 3*x^3 + 7*x^4 + 5*x^5 + x^6),x)
Output:
log(5)^2/(x^2*(x + log(- 3*x - x^2) + 1))
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-9-13 x-3 x^2\right ) \log ^2(5)+(-6-2 x) \log ^2(5) \log \left (-3 x-x^2\right )}{3 x^3+7 x^4+5 x^5+x^6+\left (6 x^3+8 x^4+2 x^5\right ) \log \left (-3 x-x^2\right )+\left (3 x^3+x^4\right ) \log ^2\left (-3 x-x^2\right )} \, dx=\frac {\mathrm {log}\left (5\right )^{2}}{x^{2} \left (\mathrm {log}\left (-x^{2}-3 x \right )+x +1\right )} \] Input:
int(((-2*x-6)*log(5)^2*log(-x^2-3*x)+(-3*x^2-13*x-9)*log(5)^2)/((x^4+3*x^3 )*log(-x^2-3*x)^2+(2*x^5+8*x^4+6*x^3)*log(-x^2-3*x)+x^6+5*x^5+7*x^4+3*x^3) ,x)
Output:
log(5)**2/(x**2*(log( - x**2 - 3*x) + x + 1))