Integrand size = 121, antiderivative size = 30 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {1}{(4+2 x) \left (5 x+\frac {(x+(-2+x) x) \log (x)}{4 x}\right )^2} \]
Time = 0.35 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8}{(2+x) (20 x+(-1+x) \log (x))^2} \]
Integrate[(32 - 656*x - 496*x^2 + (-24*x - 24*x^2)*Log[x])/(32000*x^4 + 32 000*x^5 + 8000*x^6 + (-4800*x^3 + 3600*x^5 + 1200*x^6)*Log[x] + (240*x^2 - 240*x^3 - 180*x^4 + 120*x^5 + 60*x^6)*Log[x]^2 + (-4*x + 8*x^2 - x^3 - 5* x^4 + x^5 + x^6)*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.
\(\displaystyle \int \frac {-496 x^2+\left (-24 x^2-24 x\right ) \log (x)-656 x+32}{8000 x^6+32000 x^5+32000 x^4+\left (1200 x^6+3600 x^5-4800 x^3\right ) \log (x)+\left (x^6+x^5-5 x^4-x^3+8 x^2-4 x\right ) \log ^3(x)+\left (60 x^6+120 x^5-180 x^4-240 x^3+240 x^2\right ) \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 \left (-62 x^2-82 x-3 (x+1) x \log (x)+4\right )}{x (x+2)^2 (20 x+(x-1) \log (x))^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int \frac {-62 x^2-3 (x+1) \log (x) x-82 x+4}{x (x+2)^2 (20 x-(1-x) \log (x))^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 8 \int \left (-\frac {3 (x+1)}{(x-1) (x+2)^2 (\log (x) x+20 x-\log (x))^2}-\frac {2 \left (x^2-22 x+1\right )}{(x-1) x (x+2) (\log (x) x+20 x-\log (x))^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8 \left (\frac {40}{3} \int \frac {1}{(x-1) (\log (x) x+20 x-\log (x))^3}dx+\int \frac {1}{x (\log (x) x+20 x-\log (x))^3}dx-\frac {49}{3} \int \frac {1}{(x+2) (\log (x) x+20 x-\log (x))^3}dx-\frac {2}{3} \int \frac {1}{(x-1) (\log (x) x+20 x-\log (x))^2}dx-\int \frac {1}{(x+2)^2 (\log (x) x+20 x-\log (x))^2}dx+\frac {2}{3} \int \frac {1}{(x+2) (\log (x) x+20 x-\log (x))^2}dx\right )\) |
Int[(32 - 656*x - 496*x^2 + (-24*x - 24*x^2)*Log[x])/(32000*x^4 + 32000*x^ 5 + 8000*x^6 + (-4800*x^3 + 3600*x^5 + 1200*x^6)*Log[x] + (240*x^2 - 240*x ^3 - 180*x^4 + 120*x^5 + 60*x^6)*Log[x]^2 + (-4*x + 8*x^2 - x^3 - 5*x^4 + x^5 + x^6)*Log[x]^3),x]
3.13.7.3.1 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 = 3.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {8}{\left (2+x \right ) \left (x \ln \left (x \right )-\ln \left (x \right )+20 x \right )^{2}}\) | \(22\) |
risch | \(\frac {8}{\left (2+x \right ) \left (x \ln \left (x \right )-\ln \left (x \right )+20 x \right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {8}{x^{3} \ln \left (x \right )^{2}+40 x^{3} \ln \left (x \right )+400 x^{3}+40 x^{2} \ln \left (x \right )-3 x \ln \left (x \right )^{2}+800 x^{2}-80 x \ln \left (x \right )+2 \ln \left (x \right )^{2}}\) | \(56\) |
int(((-24*x^2-24*x)*ln(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x^2-4*x) *ln(x)^3+(60*x^6+120*x^5-180*x^4-240*x^3+240*x^2)*ln(x)^2+(1200*x^6+3600*x ^5-4800*x^3)*ln(x)+8000*x^6+32000*x^5+32000*x^4),x,method=_RETURNVERBOSE)
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8}{400 \, x^{3} + {\left (x^{3} - 3 \, x + 2\right )} \log \left (x\right )^{2} + 800 \, x^{2} + 40 \, {\left (x^{3} + x^{2} - 2 \, x\right )} \log \left (x\right )} \]
integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x ^2-4*x)*log(x)^3+(60*x^6+120*x^5-180*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x ^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8}{400 x^{3} + 800 x^{2} + \left (x^{3} - 3 x + 2\right ) \log {\left (x \right )}^{2} + \left (40 x^{3} + 40 x^{2} - 80 x\right ) \log {\left (x \right )}} \]
integrate(((-24*x**2-24*x)*ln(x)-496*x**2-656*x+32)/((x**6+x**5-5*x**4-x** 3+8*x**2-4*x)*ln(x)**3+(60*x**6+120*x**5-180*x**4-240*x**3+240*x**2)*ln(x) **2+(1200*x**6+3600*x**5-4800*x**3)*ln(x)+8000*x**6+32000*x**5+32000*x**4) ,x)
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8}{400 \, x^{3} + {\left (x^{3} - 3 \, x + 2\right )} \log \left (x\right )^{2} + 800 \, x^{2} + 40 \, {\left (x^{3} + x^{2} - 2 \, x\right )} \log \left (x\right )} \]
integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x ^2-4*x)*log(x)^3+(60*x^6+120*x^5-180*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x ^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 3.80 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8 \, {\left (x^{2} - 22 \, x + 1\right )}}{x^{5} \log \left (x\right )^{2} + 40 \, x^{5} \log \left (x\right ) - 22 \, x^{4} \log \left (x\right )^{2} + 400 \, x^{5} - 840 \, x^{4} \log \left (x\right ) - 2 \, x^{3} \log \left (x\right )^{2} - 8000 \, x^{4} - 920 \, x^{3} \log \left (x\right ) + 68 \, x^{2} \log \left (x\right )^{2} - 17200 \, x^{3} + 1800 \, x^{2} \log \left (x\right ) - 47 \, x \log \left (x\right )^{2} + 800 \, x^{2} - 80 \, x \log \left (x\right ) + 2 \, \log \left (x\right )^{2}} \]
integrate(((-24*x^2-24*x)*log(x)-496*x^2-656*x+32)/((x^6+x^5-5*x^4-x^3+8*x ^2-4*x)*log(x)^3+(60*x^6+120*x^5-180*x^4-240*x^3+240*x^2)*log(x)^2+(1200*x ^6+3600*x^5-4800*x^3)*log(x)+8000*x^6+32000*x^5+32000*x^4),x, algorithm=\
8*(x^2 - 22*x + 1)/(x^5*log(x)^2 + 40*x^5*log(x) - 22*x^4*log(x)^2 + 400*x ^5 - 840*x^4*log(x) - 2*x^3*log(x)^2 - 8000*x^4 - 920*x^3*log(x) + 68*x^2* log(x)^2 - 17200*x^3 + 1800*x^2*log(x) - 47*x*log(x)^2 + 800*x^2 - 80*x*lo g(x) + 2*log(x)^2)
Time = 10.78 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {32-656 x-496 x^2+\left (-24 x-24 x^2\right ) \log (x)}{32000 x^4+32000 x^5+8000 x^6+\left (-4800 x^3+3600 x^5+1200 x^6\right ) \log (x)+\left (240 x^2-240 x^3-180 x^4+120 x^5+60 x^6\right ) \log ^2(x)+\left (-4 x+8 x^2-x^3-5 x^4+x^5+x^6\right ) \log ^3(x)} \, dx=\frac {8\,\left (x^4-20\,x^3-43\,x^2+2\,x\right )}{{\left (x+2\right )}^2\,\left (400\,x^2+{\ln \left (x\right )}^2\,{\left (x-1\right )}^2+40\,x\,\ln \left (x\right )\,\left (x-1\right )\right )\,\left (x^3-22\,x^2+x\right )} \]
int(-(656*x + log(x)*(24*x + 24*x^2) + 496*x^2 - 32)/(log(x)^2*(240*x^2 - 240*x^3 - 180*x^4 + 120*x^5 + 60*x^6) + log(x)*(3600*x^5 - 4800*x^3 + 1200 *x^6) - log(x)^3*(4*x - 8*x^2 + x^3 + 5*x^4 - x^5 - x^6) + 32000*x^4 + 320 00*x^5 + 8000*x^6),x)