Integrand size = 125, antiderivative size = 33 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=-1+\frac {x}{4 \left (\frac {x^2}{16}+\frac {\log (2)}{3}-\log \left ((-5+x) x^2\right )\right )} \]
Time = 5.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=-\frac {12 x}{-3 x^2-16 \log (2)+48 \log \left ((-5+x) x^2\right )} \]
Integrate[(-5760 + 1728*x + 180*x^2 - 36*x^3 + (-960 + 192*x)*Log[2] + (28 80 - 576*x)*Log[-5*x^2 + x^3])/(-45*x^4 + 9*x^5 + (-480*x^2 + 96*x^3)*Log[ 2] + (-1280 + 256*x)*Log[2]^2 + (1440*x^2 - 288*x^3 + (7680 - 1536*x)*Log[ 2])*Log[-5*x^2 + x^3] + (-11520 + 2304*x)*Log[-5*x^2 + x^3]^2),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.
\(\displaystyle \int \frac {-36 x^3+180 x^2+(2880-576 x) \log \left (x^3-5 x^2\right )+1728 x+(192 x-960) \log (2)-5760}{9 x^5-45 x^4+(2304 x-11520) \log ^2\left (x^3-5 x^2\right )+\left (-288 x^3+1440 x^2+(7680-1536 x) \log (2)\right ) \log \left (x^3-5 x^2\right )+\left (96 x^3-480 x^2\right ) \log (2)+(256 x-1280) \log ^2(2)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {12 \left (3 x^3-15 x^2+48 (x-5) \log \left ((x-5) x^2\right )-16 x (9+\log (2))+80 (6+\log (2))\right )}{(5-x) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 12 \int -\frac {-3 x^3+15 x^2+16 (9+\log (2)) x+48 (5-x) \log \left (-\left ((5-x) x^2\right )\right )-80 (6+\log (2))}{(5-x) \left (3 x^2-48 \log \left (-\left ((5-x) x^2\right )\right )+16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -12 \int \frac {-3 x^3+15 x^2+16 (9+\log (2)) x+48 (5-x) \log \left (-\left ((5-x) x^2\right )\right )-80 (6+\log (2))}{(5-x) \left (3 x^2-48 \log \left (-\left ((5-x) x^2\right )\right )+16 \log (2)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -12 \int \left (\frac {6 \left (x^3-5 x^2-24 x+80\right )}{(x-5) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}+\frac {1}{-3 x^2+48 \log \left ((x-5) x^2\right )-16 \log (2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -12 \left (-144 \int \frac {1}{\left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx-240 \int \frac {1}{(x-5) \left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx+6 \int \frac {x^2}{\left (3 x^2-48 \log \left ((x-5) x^2\right )+16 \log (2)\right )^2}dx+\int \frac {1}{-3 x^2+48 \log \left ((x-5) x^2\right )-16 \log (2)}dx\right )\) |
Int[(-5760 + 1728*x + 180*x^2 - 36*x^3 + (-960 + 192*x)*Log[2] + (2880 - 5 76*x)*Log[-5*x^2 + x^3])/(-45*x^4 + 9*x^5 + (-480*x^2 + 96*x^3)*Log[2] + ( -1280 + 256*x)*Log[2]^2 + (1440*x^2 - 288*x^3 + (7680 - 1536*x)*Log[2])*Lo g[-5*x^2 + x^3] + (-11520 + 2304*x)*Log[-5*x^2 + x^3]^2),x]
3.19.8.3.1 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 = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{2} \left (-5+x \right )\right )}\) | \(26\) |
norman | \(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{3}-5 x^{2}\right )}\) | \(28\) |
risch | \(\frac {12 x}{3 x^{2}+16 \ln \left (2\right )-48 \ln \left (x^{3}-5 x^{2}\right )}\) | \(28\) |
int(((-576*x+2880)*ln(x^3-5*x^2)+(192*x-960)*ln(2)-36*x^3+180*x^2+1728*x-5 760)/((2304*x-11520)*ln(x^3-5*x^2)^2+((-1536*x+7680)*ln(2)-288*x^3+1440*x^ 2)*ln(x^3-5*x^2)+(256*x-1280)*ln(2)^2+(96*x^3-480*x^2)*ln(2)+9*x^5-45*x^4) ,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x^{3} - 5 \, x^{2}\right )} \]
integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x ^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 9*x^5-45*x^4),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=- \frac {12 x}{- 3 x^{2} + 48 \log {\left (x^{3} - 5 x^{2} \right )} - 16 \log {\left (2 \right )}} \]
integrate(((-576*x+2880)*ln(x**3-5*x**2)+(192*x-960)*ln(2)-36*x**3+180*x** 2+1728*x-5760)/((2304*x-11520)*ln(x**3-5*x**2)**2+((-1536*x+7680)*ln(2)-28 8*x**3+1440*x**2)*ln(x**3-5*x**2)+(256*x-1280)*ln(2)**2+(96*x**3-480*x**2) *ln(2)+9*x**5-45*x**4),x)
Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x - 5\right ) - 96 \, \log \left (x\right )} \]
integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x ^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 9*x^5-45*x^4),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\frac {12 \, x}{3 \, x^{2} + 16 \, \log \left (2\right ) - 48 \, \log \left (x^{3} - 5 \, x^{2}\right )} \]
integrate(((-576*x+2880)*log(x^3-5*x^2)+(192*x-960)*log(2)-36*x^3+180*x^2+ 1728*x-5760)/((2304*x-11520)*log(x^3-5*x^2)^2+((-1536*x+7680)*log(2)-288*x ^3+1440*x^2)*log(x^3-5*x^2)+(256*x-1280)*log(2)^2+(96*x^3-480*x^2)*log(2)+ 9*x^5-45*x^4),x, algorithm=\
Timed out. \[ \int \frac {-5760+1728 x+180 x^2-36 x^3+(-960+192 x) \log (2)+(2880-576 x) \log \left (-5 x^2+x^3\right )}{-45 x^4+9 x^5+\left (-480 x^2+96 x^3\right ) \log (2)+(-1280+256 x) \log ^2(2)+\left (1440 x^2-288 x^3+(7680-1536 x) \log (2)\right ) \log \left (-5 x^2+x^3\right )+(-11520+2304 x) \log ^2\left (-5 x^2+x^3\right )} \, dx=\int -\frac {1728\,x+\ln \left (2\right )\,\left (192\,x-960\right )-\ln \left (x^3-5\,x^2\right )\,\left (576\,x-2880\right )+180\,x^2-36\,x^3-5760}{\ln \left (x^3-5\,x^2\right )\,\left (\ln \left (2\right )\,\left (1536\,x-7680\right )-1440\,x^2+288\,x^3\right )-{\ln \left (2\right )}^2\,\left (256\,x-1280\right )+\ln \left (2\right )\,\left (480\,x^2-96\,x^3\right )-{\ln \left (x^3-5\,x^2\right )}^2\,\left (2304\,x-11520\right )+45\,x^4-9\,x^5} \,d x \]
int(-(1728*x + log(2)*(192*x - 960) - log(x^3 - 5*x^2)*(576*x - 2880) + 18 0*x^2 - 36*x^3 - 5760)/(log(x^3 - 5*x^2)*(log(2)*(1536*x - 7680) - 1440*x^ 2 + 288*x^3) - log(2)^2*(256*x - 1280) + log(2)*(480*x^2 - 96*x^3) - log(x ^3 - 5*x^2)^2*(2304*x - 11520) + 45*x^4 - 9*x^5),x)