Integrand size = 121, antiderivative size = 29 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=-3+3 \left (5-(-3+x) x^2\right ) \log \left (5+\log \left (-x+\frac {25 x^2}{9}\right )\right ) \]
Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=-3 \left (-5+(-3+x) x^2\right ) \log \left (5+\log \left (\frac {1}{9} x (-9+25 x)\right )\right ) \]
Integrate[(-135 + 750*x - 81*x^2 + 477*x^3 - 150*x^4 + (-810*x^2 + 2655*x^ 3 - 1125*x^4 + (-162*x^2 + 531*x^3 - 225*x^4)*Log[(-9*x + 25*x^2)/9])*Log[ 5 + Log[(-9*x + 25*x^2)/9]])/(-45*x + 125*x^2 + (-9*x + 25*x^2)*Log[(-9*x + 25*x^2)/9]),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.
\(\displaystyle \int \frac {-150 x^4+477 x^3-81 x^2+\left (-1125 x^4+2655 x^3-810 x^2+\left (-225 x^4+531 x^3-162 x^2\right ) \log \left (\frac {1}{9} \left (25 x^2-9 x\right )\right )\right ) \log \left (\log \left (\frac {1}{9} \left (25 x^2-9 x\right )\right )+5\right )+750 x-135}{125 x^2+\left (25 x^2-9 x\right ) \log \left (\frac {1}{9} \left (25 x^2-9 x\right )\right )-45 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {150 x^4-477 x^3+81 x^2-\left (-1125 x^4+2655 x^3-810 x^2+\left (-225 x^4+531 x^3-162 x^2\right ) \log \left (\frac {1}{9} \left (25 x^2-9 x\right )\right )\right ) \log \left (\log \left (\frac {1}{9} \left (25 x^2-9 x\right )\right )+5\right )-750 x+135}{(9-25 x) x \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {150 x^3}{(25 x-9) \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}+\frac {477 x^2}{(25 x-9) \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}-9 (x-2) x \log \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )-\frac {81 x}{(25 x-9) \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}+\frac {750}{(25 x-9) \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}-\frac {135}{(25 x-9) x \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \int \frac {x^2}{\log \left (\frac {1}{9} x (25 x-9)\right )+5}dx-9 \int x^2 \log \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )dx+\frac {1782}{625} \int \frac {1}{\log \left (\frac {1}{9} x (25 x-9)\right )+5}dx+15 \int \frac {1}{x \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}dx+\frac {423}{25} \int \frac {x}{\log \left (\frac {1}{9} x (25 x-9)\right )+5}dx+\frac {250413}{625} \int \frac {1}{(25 x-9) \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )}dx+18 \int x \log \left (\log \left (\frac {1}{9} x (25 x-9)\right )+5\right )dx\) |
Int[(-135 + 750*x - 81*x^2 + 477*x^3 - 150*x^4 + (-810*x^2 + 2655*x^3 - 11 25*x^4 + (-162*x^2 + 531*x^3 - 225*x^4)*Log[(-9*x + 25*x^2)/9])*Log[5 + Lo g[(-9*x + 25*x^2)/9]])/(-45*x + 125*x^2 + (-9*x + 25*x^2)*Log[(-9*x + 25*x ^2)/9]),x]
3.19.73.3.1 Defintions of rubi rules used
Time = 0.49 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83
method | result | size |
parallelrisch | \(-3 \ln \left (\ln \left (\frac {25}{9} x^{2}-x \right )+5\right ) x^{3}+9 \ln \left (\ln \left (\frac {25}{9} x^{2}-x \right )+5\right ) x^{2}+15 \ln \left (\ln \left (\frac {25}{9} x^{2}-x \right )+5\right )\) | \(53\) |
int((((-225*x^4+531*x^3-162*x^2)*ln(25/9*x^2-x)-1125*x^4+2655*x^3-810*x^2) *ln(ln(25/9*x^2-x)+5)-150*x^4+477*x^3-81*x^2+750*x-135)/((25*x^2-9*x)*ln(2 5/9*x^2-x)+125*x^2-45*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=-3 \, {\left (x^{3} - 3 \, x^{2} - 5\right )} \log \left (\log \left (\frac {25}{9} \, x^{2} - x\right ) + 5\right ) \]
integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-8 10*x^2)*log(log(25/9*x^2-x)+5)-150*x^4+477*x^3-81*x^2+750*x-135)/((25*x^2- 9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm=\
Time = 0.39 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=\left (- 3 x^{3} + 9 x^{2} - \frac {54189}{312500}\right ) \log {\left (\log {\left (\frac {25 x^{2}}{9} - x \right )} + 5 \right )} + \frac {4741689 \log {\left (\log {\left (\frac {25 x^{2}}{9} - x \right )} + 5 \right )}}{312500} \]
integrate((((-225*x**4+531*x**3-162*x**2)*ln(25/9*x**2-x)-1125*x**4+2655*x **3-810*x**2)*ln(ln(25/9*x**2-x)+5)-150*x**4+477*x**3-81*x**2+750*x-135)/( (25*x**2-9*x)*ln(25/9*x**2-x)+125*x**2-45*x),x)
(-3*x**3 + 9*x**2 - 54189/312500)*log(log(25*x**2/9 - x) + 5) + 4741689*lo g(log(25*x**2/9 - x) + 5)/312500
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=-3 \, {\left (x^{3} - 3 \, x^{2} - 5\right )} \log \left (-2 \, \log \left (3\right ) + \log \left (25 \, x - 9\right ) + \log \left (x\right ) + 5\right ) \]
integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-8 10*x^2)*log(log(25/9*x^2-x)+5)-150*x^4+477*x^3-81*x^2+750*x-135)/((25*x^2- 9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=-3 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \left (\log \left (\frac {25}{9} \, x^{2} - x\right ) + 5\right ) + 15 \, \log \left (-2 \, \log \left (3\right ) + \log \left (25 \, x^{2} - 9 \, x\right ) + 5\right ) \]
integrate((((-225*x^4+531*x^3-162*x^2)*log(25/9*x^2-x)-1125*x^4+2655*x^3-8 10*x^2)*log(log(25/9*x^2-x)+5)-150*x^4+477*x^3-81*x^2+750*x-135)/((25*x^2- 9*x)*log(25/9*x^2-x)+125*x^2-45*x),x, algorithm=\
Time = 11.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-135+750 x-81 x^2+477 x^3-150 x^4+\left (-810 x^2+2655 x^3-1125 x^4+\left (-162 x^2+531 x^3-225 x^4\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right ) \log \left (5+\log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )\right )}{-45 x+125 x^2+\left (-9 x+25 x^2\right ) \log \left (\frac {1}{9} \left (-9 x+25 x^2\right )\right )} \, dx=3\,\ln \left (\ln \left (\frac {25\,x^2}{9}-x\right )+5\right )\,\left (-x^3+3\,x^2+5\right ) \]