Integrand size = 173, antiderivative size = 27 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=5 \left (-x+\log \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right )\right ) \] Output:
5*ln(x+ln((-3+x)/x/(x-9*ln(x)^2)))-5*x
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=-5 x+5 \log \left (x+\log \left (\frac {-3+x}{x \left (x-9 \log ^2(x)\right )}\right )\right ) \] Input:
Integrate[(-30*x + 20*x^2 - 20*x^3 + 5*x^4 + (270 - 90*x)*Log[x] + (135 - 135*x + 180*x^2 - 45*x^3)*Log[x]^2 + (-15*x^2 + 5*x^3 + (135*x - 45*x^2)*L og[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)])/(3*x^3 - x^4 + (-27*x^2 + 9*x ^3)*Log[x]^2 + (3*x^2 - x^3 + (-27*x + 9*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)]),x]
Output:
-5*x + 5*Log[x + Log[(-3 + x)/(x*(x - 9*Log[x]^2))]]
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.
| \(\displaystyle \int \frac {5 x^4-20 x^3+20 x^2+\left (-45 x^3+180 x^2-135 x+135\right ) \log ^2(x)+\left (5 x^3-15 x^2+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{9 x \log ^2(x)-x^2}\right )-30 x+(270-90 x) \log (x)}{-x^4+3 x^3+\left (9 x^3-27 x^2\right ) \log ^2(x)+\left (-x^3+3 x^2+\left (9 x^2-27 x\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{9 x \log ^2(x)-x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {5 x^4-20 x^3+20 x^2+\left (-45 x^3+180 x^2-135 x+135\right ) \log ^2(x)+\left (5 x^3-15 x^2+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{9 x \log ^2(x)-x^2}\right )-30 x+(270-90 x) \log (x)}{(3-x) x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {5 x^3}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {20 x^2}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {45 \left (x^3-4 x^2+3 x-3\right ) \log ^2(x)}{(x-3) x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}-\frac {20 x}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}-\frac {5 \log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )}{x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )}+\frac {30}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}+\frac {90 \log (x)}{x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -5 \int \frac {x^2}{\left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+5 \int \frac {x}{x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )}dx-5 \int \frac {1}{\left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+15 \int \frac {1}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+5 \int \frac {x}{\left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+90 \int \frac {\log (x)}{x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx-45 \int \frac {\log ^2(x)}{(x-3) \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+45 \int \frac {\log ^2(x)}{x \left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+45 \int \frac {x \log ^2(x)}{\left (x-9 \log ^2(x)\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx+45 \int \frac {\log ^2(x)}{\left (9 \log ^2(x)-x\right ) \left (x+\log \left (\frac {x-3}{x \left (x-9 \log ^2(x)\right )}\right )\right )}dx-5 x\) |
Input:
Int[(-30*x + 20*x^2 - 20*x^3 + 5*x^4 + (270 - 90*x)*Log[x] + (135 - 135*x + 180*x^2 - 45*x^3)*Log[x]^2 + (-15*x^2 + 5*x^3 + (135*x - 45*x^2)*Log[x]^ 2)*Log[(3 - x)/(-x^2 + 9*x*Log[x]^2)])/(3*x^3 - x^4 + (-27*x^2 + 9*x^3)*Lo g[x]^2 + (3*x^2 - x^3 + (-27*x + 9*x^2)*Log[x]^2)*Log[(3 - x)/(-x^2 + 9*x* Log[x]^2)]),x]
Output:
$Aborted
Time = 4.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
| method | result | size |
| parallelrisch | \(-30+5 \ln \left (x +\ln \left (-\frac {-3+x}{x \left (9 \ln \left (x \right )^{2}-x \right )}\right )\right )-5 x\) | \(32\) |
| risch | \(-5 x +5 \ln \left (\ln \left (x -9 \ln \left (x \right )^{2}\right )+\frac {i \left (\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (9 \ln \left (x \right )^{2}-x \right )}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (9 \ln \left (x \right )^{2}-x \right )}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (\frac {i}{9 \ln \left (x \right )^{2}-x}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right )-\pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{9 \ln \left (x \right )^{2}-x}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right )^{3}+\pi \,\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{9 \ln \left (x \right )^{2}-x}\right ) {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (9 \ln \left (x \right )^{2}-x \right )}\right )}^{2}-\pi {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (9 \ln \left (x \right )^{2}-x \right )}\right )}^{3}+2 i x -2 i \ln \left (x \right )+2 i \ln \left (-3+x \right )\right )}{2}\right )\) | \(332\) |
| default | \(-5 x +5 \ln \left (\ln \left (x \right )+\frac {i \pi \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right )^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )}\right )}^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}-\frac {x}{9}}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )}\right )}{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )}\right )}^{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (-3+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right ) {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )}\right )}^{2}}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (-3+x \right )}{x \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )}\right )}^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )^{2}-\frac {x}{9}}\right ) \operatorname {csgn}\left (\frac {i \left (-3+x \right )}{\ln \left (x \right )^{2}-\frac {x}{9}}\right )^{2}}{2}-i \pi +2 \ln \left (3\right )-x -\ln \left (-3+x \right )+\ln \left (\ln \left (x \right )^{2}-\frac {x}{9}\right )\right )\) | \(345\) |
Input:
int((((-45*x^2+135*x)*ln(x)^2+5*x^3-15*x^2)*ln((-x+3)/(9*x*ln(x)^2-x^2))+( -45*x^3+180*x^2-135*x+135)*ln(x)^2+(-90*x+270)*ln(x)+5*x^4-20*x^3+20*x^2-3 0*x)/(((9*x^2-27*x)*ln(x)^2-x^3+3*x^2)*ln((-x+3)/(9*x*ln(x)^2-x^2))+(9*x^3 -27*x^2)*ln(x)^2-x^4+3*x^3),x,method=_RETURNVERBOSE)
Output:
-30+5*ln(x+ln(-(-3+x)/x/(9*ln(x)^2-x)))-5*x
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=-5 \, x + 5 \, \log \left (x + \log \left (-\frac {x - 3}{9 \, x \log \left (x\right )^{2} - x^{2}}\right )\right ) \] Input:
integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2 -x^2))+(-45*x^3+180*x^2-135*x+135)*log(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^ 3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^2- x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="fricas")
Output:
-5*x + 5*log(x + log(-(x - 3)/(9*x*log(x)^2 - x^2)))
Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=- 5 x + 5 \log {\left (x + \log {\left (\frac {3 - x}{- x^{2} + 9 x \log {\left (x \right )}^{2}} \right )} \right )} \] Input:
integrate((((-45*x**2+135*x)*ln(x)**2+5*x**3-15*x**2)*ln((3-x)/(9*x*ln(x)* *2-x**2))+(-45*x**3+180*x**2-135*x+135)*ln(x)**2+(-90*x+270)*ln(x)+5*x**4- 20*x**3+20*x**2-30*x)/(((9*x**2-27*x)*ln(x)**2-x**3+3*x**2)*ln((3-x)/(9*x* ln(x)**2-x**2))+(9*x**3-27*x**2)*ln(x)**2-x**4+3*x**3),x)
Output:
-5*x + 5*log(x + log((3 - x)/(-x**2 + 9*x*log(x)**2)))
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=-5 \, x + 5 \, \log \left (-x + \log \left (-9 \, \log \left (x\right )^{2} + x\right ) - \log \left (x - 3\right ) + \log \left (x\right )\right ) \] Input:
integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2 -x^2))+(-45*x^3+180*x^2-135*x+135)*log(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^ 3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^2- x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="maxima")
Output:
-5*x + 5*log(-x + log(-9*log(x)^2 + x) - log(x - 3) + log(x))
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=-5 \, x + 5 \, \log \left (-x + \log \left (-9 \, \log \left (x\right )^{2} + x\right ) - \log \left (x - 3\right ) + \log \left (x\right )\right ) \] Input:
integrate((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2 -x^2))+(-45*x^3+180*x^2-135*x+135)*log(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^ 3+20*x^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^2- x^2))+(9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x, algorithm="giac")
Output:
-5*x + 5*log(-x + log(-9*log(x)^2 + x) - log(x - 3) + log(x))
Time = 3.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=5\,\ln \left (x+\ln \left (-\frac {x-3}{9\,x\,{\ln \left (x\right )}^2-x^2}\right )\right )-5\,x \] Input:
int((30*x + log(x)^2*(135*x - 180*x^2 + 45*x^3 - 135) + log(x)*(90*x - 270 ) - 20*x^2 + 20*x^3 - 5*x^4 - log(-(x - 3)/(9*x*log(x)^2 - x^2))*(log(x)^2 *(135*x - 45*x^2) - 15*x^2 + 5*x^3))/(log(x)^2*(27*x^2 - 9*x^3) + log(-(x - 3)/(9*x*log(x)^2 - x^2))*(log(x)^2*(27*x - 9*x^2) - 3*x^2 + x^3) - 3*x^3 + x^4),x)
Output:
5*log(x + log(-(x - 3)/(9*x*log(x)^2 - x^2))) - 5*x
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {-30 x+20 x^2-20 x^3+5 x^4+(270-90 x) \log (x)+\left (135-135 x+180 x^2-45 x^3\right ) \log ^2(x)+\left (-15 x^2+5 x^3+\left (135 x-45 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )}{3 x^3-x^4+\left (-27 x^2+9 x^3\right ) \log ^2(x)+\left (3 x^2-x^3+\left (-27 x+9 x^2\right ) \log ^2(x)\right ) \log \left (\frac {3-x}{-x^2+9 x \log ^2(x)}\right )} \, dx=5 \,\mathrm {log}\left (\mathrm {log}\left (\frac {-x +3}{9 \mathrm {log}\left (x \right )^{2} x -x^{2}}\right )+x \right )-5 x \] Input:
int((((-45*x^2+135*x)*log(x)^2+5*x^3-15*x^2)*log((3-x)/(9*x*log(x)^2-x^2)) +(-45*x^3+180*x^2-135*x+135)*log(x)^2+(-90*x+270)*log(x)+5*x^4-20*x^3+20*x ^2-30*x)/(((9*x^2-27*x)*log(x)^2-x^3+3*x^2)*log((3-x)/(9*x*log(x)^2-x^2))+ (9*x^3-27*x^2)*log(x)^2-x^4+3*x^3),x)
Output:
5*(log(log(( - x + 3)/(9*log(x)**2*x - x**2)) + x) - x)