Integrand size = 97, antiderivative size = 22 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\left (x-\frac {x}{2-\frac {x^2 \log (x)}{\log (3)}}\right )^2 \] Output:
(x-x/(-ln(x)*x^2/ln(3)+2))^2
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(22)=44\).
Time = 6.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 10.27 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\frac {x^2 \left (-x^6 \left (\log ^2(3)+\log (3) \log (9)-\log ^2(9)\right )+8 \log ^3(9) \left (\log ^2(3)-6 \log (3) \log (9)+3 \log ^2(9)\right )+4 x^2 \log ^2(9) \left (\log ^2(3)-13 \log (3) \log (9)+7 \log ^2(9)\right )+x^4 \left (-20 \log (3) \log ^2(9)+11 \log ^3(9)+\log ^2(3) \log (81)\right )-\left (x^8 \log (9)+6 x^6 \left (2 \log ^2(3)-5 \log (3) \log (9)+3 \log ^2(9)\right )+4 x^2 \log ^2(9) \left (8 \log ^2(3)-18 \log (3) \log (9)+9 \log ^2(9)\right )+4 x^4 \log (9) \left (8 \log ^2(3)-20 \log (3) \log (9)+11 \log ^2(9)\right )\right ) \log (x)+x^4 \left (x^2+\log (81)\right )^3 \log ^2(x)\right )}{\left (x^2+\log (81)\right )^3 \left (\log (9)-x^2 \log (x)\right )^2} \] Input:
Integrate[(2*x^3*Log[3]^2 - 4*x*Log[3]^3 + (-2*x^5*Log[3] + 14*x^3*Log[3]^ 2)*Log[x] - 12*x^5*Log[3]*Log[x]^2 + 2*x^7*Log[x]^3)/(-8*Log[3]^3 + 12*x^2 *Log[3]^2*Log[x] - 6*x^4*Log[3]*Log[x]^2 + x^6*Log[x]^3),x]
Output:
(x^2*(-(x^6*(Log[3]^2 + Log[3]*Log[9] - Log[9]^2)) + 8*Log[9]^3*(Log[3]^2 - 6*Log[3]*Log[9] + 3*Log[9]^2) + 4*x^2*Log[9]^2*(Log[3]^2 - 13*Log[3]*Log [9] + 7*Log[9]^2) + x^4*(-20*Log[3]*Log[9]^2 + 11*Log[9]^3 + Log[3]^2*Log[ 81]) - (x^8*Log[9] + 6*x^6*(2*Log[3]^2 - 5*Log[3]*Log[9] + 3*Log[9]^2) + 4 *x^2*Log[9]^2*(8*Log[3]^2 - 18*Log[3]*Log[9] + 9*Log[9]^2) + 4*x^4*Log[9]* (8*Log[3]^2 - 20*Log[3]*Log[9] + 11*Log[9]^2))*Log[x] + x^4*(x^2 + Log[81] )^3*Log[x]^2))/((x^2 + Log[81])^3*(Log[9] - x^2*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.
\(\displaystyle \int \frac {2 x^7 \log ^3(x)-12 x^5 \log (3) \log ^2(x)+2 x^3 \log ^2(3)+\left (14 x^3 \log ^2(3)-2 x^5 \log (3)\right ) \log (x)-4 x \log ^3(3)}{x^6 \log ^3(x)-6 x^4 \log (3) \log ^2(x)+12 x^2 \log ^2(3) \log (x)-8 \log ^3(3)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (x^7 \left (-\log ^3(x)\right )+6 x^5 \log (3) \log ^2(x)+x^5 \log (3) \log (x)-7 x^3 \log ^2(3) \log (x)-x^3 \log ^2(3)+x \log ^2(3) \log (9)\right )}{\left (\log (9)-x^2 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\log ^3(x) x^7-6 \log (3) \log ^2(x) x^5-\log (3) \log (x) x^5+7 \log ^2(3) \log (x) x^3+\log ^2(3) x^3-\log ^2(3) \log (9) x}{\left (\log (9)-x^2 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\log ^3(x) x^7-6 \log (3) \log ^2(x) x^5-\log (3) \log (x) x^5+7 \log ^2(3) \log (x) x^3+\log ^2(3) x^3-\log ^2(3) \log (9) x}{\left (\log (9)-x^2 \log (x)\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {\left (\log (3) x^2-3 \log ^2(9)+12 \log (3) \log (9)-7 \log ^2(3)\right ) x}{\left (\log (9)-x^2 \log (x)\right )^2}+\frac {\left (\log (9) \left (6 \log ^2(3)-6 \log (3) \log (9)+\log ^2(9)\right )-x^2 \log ^2(3)\right ) x}{\left (\log (9)-x^2 \log (x)\right )^3}-x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (\log (9) \left (6 \log ^2(3)+\log ^2(9)-6 \log (3) \log (9)\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^3}dx-\left (7 \log ^2(3)+3 \log ^2(9)-12 \log (3) \log (9)\right ) \int \frac {x}{\left (\log (9)-x^2 \log (x)\right )^2}dx-\log ^2(3) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^3}dx+\log (3) \int \frac {x^3}{\left (\log (9)-x^2 \log (x)\right )^2}dx-\frac {x^2}{2}\right )\) |
Input:
Int[(2*x^3*Log[3]^2 - 4*x*Log[3]^3 + (-2*x^5*Log[3] + 14*x^3*Log[3]^2)*Log [x] - 12*x^5*Log[3]*Log[x]^2 + 2*x^7*Log[x]^3)/(-8*Log[3]^3 + 12*x^2*Log[3 ]^2*Log[x] - 6*x^4*Log[3]*Log[x]^2 + x^6*Log[x]^3),x]
Output:
$Aborted
Time = 1.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73
method | result | size |
risch | \(x^{2}-\frac {\left (-2 x^{2} \ln \left (x \right )+3 \ln \left (3\right )\right ) \ln \left (3\right ) x^{2}}{\left (-x^{2} \ln \left (x \right )+2 \ln \left (3\right )\right )^{2}}\) | \(38\) |
default | \(\frac {2 \ln \left (3\right )}{\ln \left (x \right )}+x^{2}-\frac {\left (-5 x^{2} \ln \left (x \right )+8 \ln \left (3\right )\right ) \ln \left (3\right )^{2}}{\ln \left (x \right ) \left (-x^{2} \ln \left (x \right )+2 \ln \left (3\right )\right )^{2}}\) | \(49\) |
parallelrisch | \(\frac {x^{6} \ln \left (x \right )^{2}-2 \ln \left (x \right ) \ln \left (3\right ) x^{4}+x^{2} \ln \left (3\right )^{2}}{x^{4} \ln \left (x \right )^{2}-4 \ln \left (x \right ) \ln \left (3\right ) x^{2}+4 \ln \left (3\right )^{2}}\) | \(54\) |
Input:
int((2*x^7*ln(x)^3-12*x^5*ln(3)*ln(x)^2+(14*x^3*ln(3)^2-2*x^5*ln(3))*ln(x) -4*x*ln(3)^3+2*x^3*ln(3)^2)/(x^6*ln(x)^3-6*x^4*ln(3)*ln(x)^2+12*x^2*ln(3)^ 2*ln(x)-8*ln(3)^3),x,method=_RETURNVERBOSE)
Output:
x^2-(-2*x^2*ln(x)+3*ln(3))*ln(3)*x^2/(-x^2*ln(x)+2*ln(3))^2
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\frac {x^{6} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (3\right ) \log \left (x\right ) + x^{2} \log \left (3\right )^{2}}{x^{4} \log \left (x\right )^{2} - 4 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (3\right )^{2}} \] Input:
integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*lo g(3))*log(x)-4*x*log(3)^3+2*x^3*log(3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x )^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="fricas")
Output:
(x^6*log(x)^2 - 2*x^4*log(3)*log(x) + x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2* log(3)*log(x) + 4*log(3)^2)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (15) = 30\).
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=x^{2} + \frac {2 x^{4} \log {\left (3 \right )} \log {\left (x \right )} - 3 x^{2} \log {\left (3 \right )}^{2}}{x^{4} \log {\left (x \right )}^{2} - 4 x^{2} \log {\left (3 \right )} \log {\left (x \right )} + 4 \log {\left (3 \right )}^{2}} \] Input:
integrate((2*x**7*ln(x)**3-12*x**5*ln(3)*ln(x)**2+(14*x**3*ln(3)**2-2*x**5 *ln(3))*ln(x)-4*x*ln(3)**3+2*x**3*ln(3)**2)/(x**6*ln(x)**3-6*x**4*ln(3)*ln (x)**2+12*x**2*ln(3)**2*ln(x)-8*ln(3)**3),x)
Output:
x**2 + (2*x**4*log(3)*log(x) - 3*x**2*log(3)**2)/(x**4*log(x)**2 - 4*x**2* log(3)*log(x) + 4*log(3)**2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\frac {x^{6} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (3\right ) \log \left (x\right ) + x^{2} \log \left (3\right )^{2}}{x^{4} \log \left (x\right )^{2} - 4 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (3\right )^{2}} \] Input:
integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*lo g(3))*log(x)-4*x*log(3)^3+2*x^3*log(3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x )^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="maxima")
Output:
(x^6*log(x)^2 - 2*x^4*log(3)*log(x) + x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2* log(3)*log(x) + 4*log(3)^2)
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.27 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=x^{2} + \frac {2 \, x^{4} \log \left (3\right ) \log \left (x\right ) - 3 \, x^{2} \log \left (3\right )^{2}}{x^{4} \log \left (x\right )^{2} - 4 \, x^{2} \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (3\right )^{2}} \] Input:
integrate((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*lo g(3))*log(x)-4*x*log(3)^3+2*x^3*log(3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x )^2+12*x^2*log(3)^2*log(x)-8*log(3)^3),x, algorithm="giac")
Output:
x^2 + (2*x^4*log(3)*log(x) - 3*x^2*log(3)^2)/(x^4*log(x)^2 - 4*x^2*log(3)* log(x) + 4*log(3)^2)
Time = 4.25 (sec) , antiderivative size = 229, normalized size of antiderivative = 10.41 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\frac {\frac {52\,x^2\,{\ln \left (3\right )}^3+11\,x^4\,{\ln \left (3\right )}^2-8\,{\ln \left (3\right )}^3\,\ln \left (9\right )+x^6\,\ln \left (3\right )+64\,{\ln \left (3\right )}^4-4\,x^2\,{\ln \left (3\right )}^2\,\ln \left (9\right )}{{\left (x^2+4\,\ln \left (3\right )\right )}^3}-\frac {2\,x^4\,{\ln \left (3\right )}^2\,\ln \left (x\right )}{{\left (x^2+4\,\ln \left (3\right )\right )}^3}}{\ln \left (x\right )-\frac {2\,\ln \left (3\right )}{x^2}}+x^2-\frac {\frac {{\ln \left (3\right )}^2\,x^2-{\ln \left (3\right )}^2\,\ln \left (9\right )+8\,{\ln \left (3\right )}^3}{x^2\,\left (x^2+4\,\ln \left (3\right )\right )}-\frac {\ln \left (x\right )\,\left (\ln \left (3\right )\,x^2+5\,{\ln \left (3\right )}^2\right )}{x^2+4\,\ln \left (3\right )}}{\frac {4\,{\ln \left (3\right )}^2}{x^4}+{\ln \left (x\right )}^2-\frac {4\,\ln \left (3\right )\,\ln \left (x\right )}{x^2}}+\frac {2\,x^4\,{\ln \left (3\right )}^2}{x^6+12\,\ln \left (3\right )\,x^4+48\,{\ln \left (3\right )}^2\,x^2+64\,{\ln \left (3\right )}^3} \] Input:
int((2*x^3*log(3)^2 + log(x)*(14*x^3*log(3)^2 - 2*x^5*log(3)) + 2*x^7*log( x)^3 - 4*x*log(3)^3 - 12*x^5*log(3)*log(x)^2)/(x^6*log(x)^3 - 8*log(3)^3 + 12*x^2*log(3)^2*log(x) - 6*x^4*log(3)*log(x)^2),x)
Output:
((52*x^2*log(3)^3 + 11*x^4*log(3)^2 - 8*log(3)^3*log(9) + x^6*log(3) + 64* log(3)^4 - 4*x^2*log(3)^2*log(9))/(4*log(3) + x^2)^3 - (2*x^4*log(3)^2*log (x))/(4*log(3) + x^2)^3)/(log(x) - (2*log(3))/x^2) + x^2 - ((x^2*log(3)^2 - log(3)^2*log(9) + 8*log(3)^3)/(x^2*(4*log(3) + x^2)) - (log(x)*(x^2*log( 3) + 5*log(3)^2))/(4*log(3) + x^2))/((4*log(3)^2)/x^4 + log(x)^2 - (4*log( 3)*log(x))/x^2) + (2*x^4*log(3)^2)/(48*x^2*log(3)^2 + 12*x^4*log(3) + 64*l og(3)^3 + x^6)
Time = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {2 x^3 \log ^2(3)-4 x \log ^3(3)+\left (-2 x^5 \log (3)+14 x^3 \log ^2(3)\right ) \log (x)-12 x^5 \log (3) \log ^2(x)+2 x^7 \log ^3(x)}{-8 \log ^3(3)+12 x^2 \log ^2(3) \log (x)-6 x^4 \log (3) \log ^2(x)+x^6 \log ^3(x)} \, dx=\frac {x^{2} \left (\mathrm {log}\left (x \right )^{2} x^{4}-2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{2}+\mathrm {log}\left (3\right )^{2}\right )}{\mathrm {log}\left (x \right )^{2} x^{4}-4 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x^{2}+4 \mathrm {log}\left (3\right )^{2}} \] Input:
int((2*x^7*log(x)^3-12*x^5*log(3)*log(x)^2+(14*x^3*log(3)^2-2*x^5*log(3))* log(x)-4*x*log(3)^3+2*x^3*log(3)^2)/(x^6*log(x)^3-6*x^4*log(3)*log(x)^2+12 *x^2*log(3)^2*log(x)-8*log(3)^3),x)
Output:
(x**2*(log(x)**2*x**4 - 2*log(x)*log(3)*x**2 + log(3)**2))/(log(x)**2*x**4 - 4*log(x)*log(3)*x**2 + 4*log(3)**2)