Integrand size = 107, antiderivative size = 28 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=x^2-\left (\frac {1+x}{3}+\log (5)\right )^2+\frac {x^2}{(x+\log (x))^2} \] Output:
x^2/(x+ln(x))^2+x^2-(1/3*x+1/3+ln(5))^2
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {1}{9} x \left (-2 (1+\log (125))+x \left (8+\frac {9}{(x+\log (x))^2}\right )\right ) \] Input:
Integrate[(-18*x - 2*x^3 + 16*x^4 - 6*x^3*Log[5] + (18*x - 6*x^2 + 48*x^3 - 18*x^2*Log[5])*Log[x] + (-6*x + 48*x^2 - 18*x*Log[5])*Log[x]^2 + (-2 + 1 6*x - 6*Log[5])*Log[x]^3)/(9*x^3 + 27*x^2*Log[x] + 27*x*Log[x]^2 + 9*Log[x ]^3),x]
Output:
(x*(-2*(1 + Log[125]) + x*(8 + 9/(x + Log[x])^2)))/9
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.
\(\displaystyle \int \frac {16 x^4-2 x^3-6 x^3 \log (5)+\left (48 x^2-6 x-18 x \log (5)\right ) \log ^2(x)+\left (48 x^3-6 x^2-18 x^2 \log (5)+18 x\right ) \log (x)-18 x+(16 x-2-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+9 \log ^3(x)+27 x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {16 x^4+x^3 (-2-6 \log (5))+\left (48 x^2-6 x-18 x \log (5)\right ) \log ^2(x)+\left (48 x^3-6 x^2-18 x^2 \log (5)+18 x\right ) \log (x)-18 x+(16 x-2-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+9 \log ^3(x)+27 x \log ^2(x)}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {16 x^4+x^3 (-2-6 \log (5))+\left (48 x^2-6 x-18 x \log (5)\right ) \log ^2(x)+\left (48 x^3-6 x^2-18 x^2 \log (5)+18 x\right ) \log (x)-18 x+(16 x-2-6 \log (5)) \log ^3(x)}{9 (x+\log (x))^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {2 \left (-8 x^4+(1+\log (125)) x^3+9 x+(-8 x+3 \log (5)+1) \log ^3(x)+3 \left (-8 x^2+3 \log (5) x+x\right ) \log ^2(x)-3 \left (8 x^3-3 \log (5) x^2-x^2+3 x\right ) \log (x)\right )}{(x+\log (x))^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2}{9} \int \frac {-8 x^4+(1+\log (125)) x^3+9 x+(-8 x+\log (125)+1) \log ^3(x)+3 \left (-8 x^2+3 \log (5) x+x\right ) \log ^2(x)-3 \left (8 x^3-3 \log (5) x^2-x^2+3 x\right ) \log (x)}{(x+\log (x))^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {2}{9} \int \left (-\frac {9 x}{(x+\log (x))^2}+\frac {9 (x+1) x}{(x+\log (x))^3}-8 x+\log (125)+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{9} \left (9 \int \frac {x^2}{(x+\log (x))^3}dx+9 \int \frac {x}{(x+\log (x))^3}dx-9 \int \frac {x}{(x+\log (x))^2}dx-4 x^2+x (1+\log (125))\right )\) |
Input:
Int[(-18*x - 2*x^3 + 16*x^4 - 6*x^3*Log[5] + (18*x - 6*x^2 + 48*x^3 - 18*x ^2*Log[5])*Log[x] + (-6*x + 48*x^2 - 18*x*Log[5])*Log[x]^2 + (-2 + 16*x - 6*Log[5])*Log[x]^3)/(9*x^3 + 27*x^2*Log[x] + 27*x*Log[x]^2 + 9*Log[x]^3),x ]
Output:
$Aborted
Time = 16.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {2 x \ln \left (5\right )}{3}+\frac {8 x^{2}}{9}-\frac {2 x}{9}+\frac {x^{2}}{\left (x +\ln \left (x \right )\right )^{2}}\) | \(25\) |
default | \(\frac {x^{2}-\frac {2 x^{3}}{9}+\frac {8 x^{4}}{9}-\frac {2 x \ln \left (x \right )^{2}}{9}-\frac {4 x^{2} \ln \left (x \right )}{9}+\frac {8 x^{2} \ln \left (x \right )^{2}}{9}+\frac {16 x^{3} \ln \left (x \right )}{9}}{\left (x +\ln \left (x \right )\right )^{2}}-\frac {2 x \ln \left (5\right )}{3}\) | \(61\) |
norman | \(\frac {-\ln \left (x \right )^{2}+\left (-\frac {2}{9}-\frac {2 \ln \left (5\right )}{3}\right ) x^{3}-2 x \ln \left (x \right )+\left (-\frac {4}{9}-\frac {4 \ln \left (5\right )}{3}\right ) x^{2} \ln \left (x \right )+\left (-\frac {2}{9}-\frac {2 \ln \left (5\right )}{3}\right ) x \ln \left (x \right )^{2}+\frac {8 x^{4}}{9}+\frac {8 x^{2} \ln \left (x \right )^{2}}{9}+\frac {16 x^{3} \ln \left (x \right )}{9}}{\left (x +\ln \left (x \right )\right )^{2}}\) | \(75\) |
parallelrisch | \(\frac {-6 x^{3} \ln \left (5\right )-12 \ln \left (x \right ) \ln \left (5\right ) x^{2}-6 x \ln \left (x \right )^{2} \ln \left (5\right )+8 x^{4}+16 x^{3} \ln \left (x \right )+8 x^{2} \ln \left (x \right )^{2}-2 x^{3}-4 x^{2} \ln \left (x \right )-2 x \ln \left (x \right )^{2}+9 x^{2}}{9 x^{2}+18 x \ln \left (x \right )+9 \ln \left (x \right )^{2}}\) | \(89\) |
Input:
int(((-6*ln(5)+16*x-2)*ln(x)^3+(-18*x*ln(5)+48*x^2-6*x)*ln(x)^2+(-18*x^2*l n(5)+48*x^3-6*x^2+18*x)*ln(x)-6*x^3*ln(5)+16*x^4-2*x^3-18*x)/(9*ln(x)^3+27 *x*ln(x)^2+27*x^2*ln(x)+9*x^3),x,method=_RETURNVERBOSE)
Output:
-2/3*x*ln(5)+8/9*x^2-2/9*x+x^2/(x+ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {8 \, x^{4} - 6 \, x^{3} \log \left (5\right ) - 2 \, x^{3} + 2 \, {\left (4 \, x^{2} - 3 \, x \log \left (5\right ) - x\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 4 \, {\left (4 \, x^{3} - 3 \, x^{2} \log \left (5\right ) - x^{2}\right )} \log \left (x\right )}{9 \, {\left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:
integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+ (-18*x^2*log(5)+48*x^3-6*x^2+18*x)*log(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/ (9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="fricas")
Output:
1/9*(8*x^4 - 6*x^3*log(5) - 2*x^3 + 2*(4*x^2 - 3*x*log(5) - x)*log(x)^2 + 9*x^2 + 4*(4*x^3 - 3*x^2*log(5) - x^2)*log(x))/(x^2 + 2*x*log(x) + log(x)^ 2)
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {8 x^{2}}{9} + \frac {x^{2}}{x^{2} + 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} + x \left (- \frac {2 \log {\left (5 \right )}}{3} - \frac {2}{9}\right ) \] Input:
integrate(((-6*ln(5)+16*x-2)*ln(x)**3+(-18*x*ln(5)+48*x**2-6*x)*ln(x)**2+( -18*x**2*ln(5)+48*x**3-6*x**2+18*x)*ln(x)-6*x**3*ln(5)+16*x**4-2*x**3-18*x )/(9*ln(x)**3+27*x*ln(x)**2+27*x**2*ln(x)+9*x**3),x)
Output:
8*x**2/9 + x**2/(x**2 + 2*x*log(x) + log(x)**2) + x*(-2*log(5)/3 - 2/9)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (25) = 50\).
Time = 0.15 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {8 \, x^{4} - 2 \, x^{3} {\left (3 \, \log \left (5\right ) + 1\right )} + 2 \, {\left (4 \, x^{2} - x {\left (3 \, \log \left (5\right ) + 1\right )}\right )} \log \left (x\right )^{2} + 9 \, x^{2} + 4 \, {\left (4 \, x^{3} - x^{2} {\left (3 \, \log \left (5\right ) + 1\right )}\right )} \log \left (x\right )}{9 \, {\left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:
integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+ (-18*x^2*log(5)+48*x^3-6*x^2+18*x)*log(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/ (9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="maxima")
Output:
1/9*(8*x^4 - 2*x^3*(3*log(5) + 1) + 2*(4*x^2 - x*(3*log(5) + 1))*log(x)^2 + 9*x^2 + 4*(4*x^3 - x^2*(3*log(5) + 1))*log(x))/(x^2 + 2*x*log(x) + log(x )^2)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {8}{9} \, x^{2} - \frac {2}{9} \, x {\left (3 \, \log \left (5\right ) + 1\right )} + \frac {x^{3} + x^{2}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:
integrate(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+ (-18*x^2*log(5)+48*x^3-6*x^2+18*x)*log(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/ (9*log(x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x, algorithm="giac")
Output:
8/9*x^2 - 2/9*x*(3*log(5) + 1) + (x^3 + x^2)/(x^3 + 2*x^2*log(x) + x*log(x )^2 + x^2 + 2*x*log(x) + log(x)^2)
Time = 4.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.18 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {8\,x^5+16\,x^4\,\ln \left (x\right )+\left (-6\,\ln \left (5\right )-2\right )\,x^4+8\,x^3\,{\ln \left (x\right )}^2+\left (-12\,\ln \left (5\right )-4\right )\,x^3\,\ln \left (x\right )+9\,x^3+\left (-6\,\ln \left (5\right )-2\right )\,x^2\,{\ln \left (x\right )}^2}{9\,x^3+18\,x^2\,\ln \left (x\right )+9\,x\,{\ln \left (x\right )}^2} \] Input:
int(-(18*x + log(x)^2*(6*x + 18*x*log(5) - 48*x^2) - log(x)*(18*x - 18*x^2 *log(5) - 6*x^2 + 48*x^3) + 6*x^3*log(5) + log(x)^3*(6*log(5) - 16*x + 2) + 2*x^3 - 16*x^4)/(27*x*log(x)^2 + 27*x^2*log(x) + 9*log(x)^3 + 9*x^3),x)
Output:
(16*x^4*log(x) - x^4*(6*log(5) + 2) + 8*x^3*log(x)^2 + 9*x^3 + 8*x^5 - x^3 *log(x)*(12*log(5) + 4) - x^2*log(x)^2*(6*log(5) + 2))/(9*x*log(x)^2 + 18* x^2*log(x) + 9*x^3)
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {-18 x-2 x^3+16 x^4-6 x^3 \log (5)+\left (18 x-6 x^2+48 x^3-18 x^2 \log (5)\right ) \log (x)+\left (-6 x+48 x^2-18 x \log (5)\right ) \log ^2(x)+(-2+16 x-6 \log (5)) \log ^3(x)}{9 x^3+27 x^2 \log (x)+27 x \log ^2(x)+9 \log ^3(x)} \, dx=\frac {x \left (-6 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (5\right )+8 \mathrm {log}\left (x \right )^{2} x -2 \mathrm {log}\left (x \right )^{2}-12 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (5\right ) x +16 \,\mathrm {log}\left (x \right ) x^{2}-4 \,\mathrm {log}\left (x \right ) x -6 \,\mathrm {log}\left (5\right ) x^{2}+8 x^{3}-2 x^{2}+9 x \right )}{9 \mathrm {log}\left (x \right )^{2}+18 \,\mathrm {log}\left (x \right ) x +9 x^{2}} \] Input:
int(((-6*log(5)+16*x-2)*log(x)^3+(-18*x*log(5)+48*x^2-6*x)*log(x)^2+(-18*x ^2*log(5)+48*x^3-6*x^2+18*x)*log(x)-6*x^3*log(5)+16*x^4-2*x^3-18*x)/(9*log (x)^3+27*x*log(x)^2+27*x^2*log(x)+9*x^3),x)
Output:
(x*( - 6*log(x)**2*log(5) + 8*log(x)**2*x - 2*log(x)**2 - 12*log(x)*log(5) *x + 16*log(x)*x**2 - 4*log(x)*x - 6*log(5)*x**2 + 8*x**3 - 2*x**2 + 9*x)) /(9*(log(x)**2 + 2*log(x)*x + x**2))