Integrand size = 125, antiderivative size = 23 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {64 (2+x)}{x^2 \left (\log \left (\frac {1}{3} (-3+x)\right )-\log (x)\right )^2} \]
Time = 5.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=-\frac {64 (-2-x)}{x^2 \left (-\log \left (\frac {1}{3} (-3+x)\right )+\log (x)\right )^2} \]
Integrate[(768 + 384*x + (-768 + 64*x + 64*x^2)*Log[(-3 + x)/3] + (768 - 6 4*x - 64*x^2)*Log[x])/((3*x^3 - x^4)*Log[(-3 + x)/3]^3 + (-9*x^3 + 3*x^4)* Log[(-3 + x)/3]^2*Log[x] + (9*x^3 - 3*x^4)*Log[(-3 + x)/3]*Log[x]^2 + (-3* x^3 + x^4)*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.
\(\displaystyle \int \frac {\left (64 x^2+64 x-768\right ) \log \left (\frac {x-3}{3}\right )+\left (-64 x^2-64 x+768\right ) \log (x)+384 x+768}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {x-3}{3}\right )+\left (x^4-3 x^3\right ) \log ^3(x)+\left (3 x^4-9 x^3\right ) \log (x) \log ^2\left (\frac {x-3}{3}\right )+\left (9 x^3-3 x^4\right ) \log ^2(x) \log \left (\frac {x-3}{3}\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {64 \left (\left (x^2+x-12\right ) \log \left (\frac {x-3}{3}\right )-\left (x^2+x-12\right ) \log (x)+6 (x+2)\right )}{(3-x) x^3 \left (\log \left (\frac {x-3}{3}\right )-\log (x)\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 64 \int \frac {6 (x+2)-\left (-x^2-x+12\right ) \log \left (\frac {x-3}{3}\right )+\left (-x^2-x+12\right ) \log (x)}{(3-x) x^3 \left (\log \left (\frac {x-3}{3}\right )-\log (x)\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 64 \int \left (\frac {-x-4}{x^3 \left (\log \left (\frac {x-3}{3}\right )-\log (x)\right )^2}-\frac {6 (x+2)}{(x-3) x^3 (\log (x-3)-\log (3 x))^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 64 \left (-4 \int \frac {1}{x^3 \left (\log \left (\frac {x-3}{3}\right )-\log (x)\right )^2}dx+4 \int \frac {1}{x^3 (\log (x-3)-\log (3 x))^3}dx-\int \frac {1}{x^2 \left (\log \left (\frac {x-3}{3}\right )-\log (x)\right )^2}dx+\frac {10}{3} \int \frac {1}{x^2 (\log (x-3)-\log (3 x))^3}dx-\frac {10}{9} \int \frac {1}{(x-3) (\log (x-3)-\log (3 x))^3}dx+\frac {10}{9} \int \frac {1}{x (\log (x-3)-\log (3 x))^3}dx\right )\) |
Int[(768 + 384*x + (-768 + 64*x + 64*x^2)*Log[(-3 + x)/3] + (768 - 64*x - 64*x^2)*Log[x])/((3*x^3 - x^4)*Log[(-3 + x)/3]^3 + (-9*x^3 + 3*x^4)*Log[(- 3 + x)/3]^2*Log[x] + (9*x^3 - 3*x^4)*Log[(-3 + x)/3]*Log[x]^2 + (-3*x^3 + x^4)*Log[x]^3),x]
3.3.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.65 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {64 x +128}{x^{2} \left (\ln \left (3\right )-\ln \left (-3+x \right )+\ln \left (x \right )\right )^{2}}\) | \(22\) |
risch | \(\frac {64 x +128}{x^{2} \left (\ln \left (x \right )-\ln \left (\frac {x}{3}-1\right )\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {1536+768 x}{12 x^{2} \left (\ln \left (x \right )^{2}-2 \ln \left (\frac {x}{3}-1\right ) \ln \left (x \right )+\ln \left (\frac {x}{3}-1\right )^{2}\right )}\) | \(36\) |
int(((-64*x^2-64*x+768)*ln(x)+(64*x^2+64*x-768)*ln(1/3*x-1)+384*x+768)/((x ^4-3*x^3)*ln(x)^3+(-3*x^4+9*x^3)*ln(1/3*x-1)*ln(x)^2+(3*x^4-9*x^3)*ln(1/3* x-1)^2*ln(x)+(-x^4+3*x^3)*ln(1/3*x-1)^3),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {64 \, {\left (x + 2\right )}}{x^{2} \log \left (x\right )^{2} - 2 \, x^{2} \log \left (x\right ) \log \left (\frac {1}{3} \, x - 1\right ) + x^{2} \log \left (\frac {1}{3} \, x - 1\right )^{2}} \]
integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+ 768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x ^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {64 x + 128}{x^{2} \log {\left (x \right )}^{2} - 2 x^{2} \log {\left (x \right )} \log {\left (\frac {x}{3} - 1 \right )} + x^{2} \log {\left (\frac {x}{3} - 1 \right )}^{2}} \]
integrate(((-64*x**2-64*x+768)*ln(x)+(64*x**2+64*x-768)*ln(1/3*x-1)+384*x+ 768)/((x**4-3*x**3)*ln(x)**3+(-3*x**4+9*x**3)*ln(1/3*x-1)*ln(x)**2+(3*x**4 -9*x**3)*ln(1/3*x-1)**2*ln(x)+(-x**4+3*x**3)*ln(1/3*x-1)**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (21) = 42\).
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {64 \, {\left (x + 2\right )}}{x^{2} \log \left (3\right )^{2} + x^{2} \log \left (x - 3\right )^{2} + 2 \, x^{2} \log \left (3\right ) \log \left (x\right ) + x^{2} \log \left (x\right )^{2} - 2 \, {\left (x^{2} \log \left (3\right ) + x^{2} \log \left (x\right )\right )} \log \left (x - 3\right )} \]
integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+ 768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x ^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm=\
64*(x + 2)/(x^2*log(3)^2 + x^2*log(x - 3)^2 + 2*x^2*log(3)*log(x) + x^2*lo g(x)^2 - 2*(x^2*log(3) + x^2*log(x))*log(x - 3))
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.70 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {64 \, {\left (x + 2\right )}}{{\left (x - 3\right )}^{2} \log \left (x\right )^{2} - 2 \, {\left (x - 3\right )}^{2} \log \left (x\right ) \log \left (\frac {1}{3} \, x - 1\right ) + {\left (x - 3\right )}^{2} \log \left (\frac {1}{3} \, x - 1\right )^{2} + 6 \, {\left (x - 3\right )} \log \left (x\right )^{2} - 12 \, {\left (x - 3\right )} \log \left (x\right ) \log \left (\frac {1}{3} \, x - 1\right ) + 6 \, {\left (x - 3\right )} \log \left (\frac {1}{3} \, x - 1\right )^{2} + 9 \, \log \left (x\right )^{2} - 18 \, \log \left (x\right ) \log \left (\frac {1}{3} \, x - 1\right ) + 9 \, \log \left (\frac {1}{3} \, x - 1\right )^{2}} \]
integrate(((-64*x^2-64*x+768)*log(x)+(64*x^2+64*x-768)*log(1/3*x-1)+384*x+ 768)/((x^4-3*x^3)*log(x)^3+(-3*x^4+9*x^3)*log(1/3*x-1)*log(x)^2+(3*x^4-9*x ^3)*log(1/3*x-1)^2*log(x)+(-x^4+3*x^3)*log(1/3*x-1)^3),x, algorithm=\
64*(x + 2)/((x - 3)^2*log(x)^2 - 2*(x - 3)^2*log(x)*log(1/3*x - 1) + (x - 3)^2*log(1/3*x - 1)^2 + 6*(x - 3)*log(x)^2 - 12*(x - 3)*log(x)*log(1/3*x - 1) + 6*(x - 3)*log(1/3*x - 1)^2 + 9*log(x)^2 - 18*log(x)*log(1/3*x - 1) + 9*log(1/3*x - 1)^2)
Time = 8.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {768+384 x+\left (-768+64 x+64 x^2\right ) \log \left (\frac {1}{3} (-3+x)\right )+\left (768-64 x-64 x^2\right ) \log (x)}{\left (3 x^3-x^4\right ) \log ^3\left (\frac {1}{3} (-3+x)\right )+\left (-9 x^3+3 x^4\right ) \log ^2\left (\frac {1}{3} (-3+x)\right ) \log (x)+\left (9 x^3-3 x^4\right ) \log \left (\frac {1}{3} (-3+x)\right ) \log ^2(x)+\left (-3 x^3+x^4\right ) \log ^3(x)} \, dx=\frac {32\,\left (-x^2\,{\ln \left (\frac {x}{3}-1\right )}^2+2\,x^2\,\ln \left (\frac {x}{3}-1\right )\,\ln \left (x\right )-x^2\,{\ln \left (x\right )}^2+18\,x+36\right )}{9\,x^2\,{\left (\ln \left (\frac {x}{3}-1\right )-\ln \left (x\right )\right )}^2} \]
int((384*x + log(x/3 - 1)*(64*x + 64*x^2 - 768) - log(x)*(64*x + 64*x^2 - 768) + 768)/(log(x/3 - 1)^3*(3*x^3 - x^4) - log(x)^3*(3*x^3 - x^4) + log(x /3 - 1)*log(x)^2*(9*x^3 - 3*x^4) - log(x/3 - 1)^2*log(x)*(9*x^3 - 3*x^4)), x)