Integrand size = 37, antiderivative size = 265 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (-8+13 x^3\right )}{40 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2\ 2^{2/3} \sqrt [3]{-1+x^3}}\right )}{8 \sqrt [3]{2}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {\log \left (-3 x+6^{2/3} \sqrt [3]{-1+x^3}\right )}{8 \sqrt [3]{6}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}}+\frac {\log \left (3 x^2+6^{2/3} x \sqrt [3]{-1+x^3}+2 \sqrt [3]{6} \left (-1+x^3\right )^{2/3}\right )}{16 \sqrt [3]{6}} \]
1/40*(x^3-1)^(2/3)*(13*x^3-8)/x^5-1/12*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/ 3)*(x^3-1)^(1/3)))*2^(1/3)+1/16*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(2 /3)*(x^3-1)^(1/3)))*2^(2/3)+1/12*ln(-x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)-1/48 *ln(-3*x+6^(2/3)*(x^3-1)^(1/3))*6^(2/3)-1/24*ln(x^2+2^(1/3)*x*(x^3-1)^(1/3 )+2^(2/3)*(x^3-1)^(2/3))*2^(1/3)+1/96*ln(3*x^2+6^(2/3)*x*(x^3-1)^(1/3)+2*6 ^(1/3)*(x^3-1)^(2/3))*6^(2/3)
Time = 0.66 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\frac {1}{480} \left (\frac {12 \left (-1+x^3\right )^{2/3} \left (-8+13 x^3\right )}{x^5}-40 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+30\ 2^{2/3} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2\ 2^{2/3} \sqrt [3]{-1+x^3}}\right )+40 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-10\ 6^{2/3} \log \left (-3 x+6^{2/3} \sqrt [3]{-1+x^3}\right )-20 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )+5\ 6^{2/3} \log \left (3 x^2+6^{2/3} x \sqrt [3]{-1+x^3}+2 \sqrt [3]{6} \left (-1+x^3\right )^{2/3}\right )\right ) \]
((12*(-1 + x^3)^(2/3)*(-8 + 13*x^3))/x^5 - 40*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt [3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))] + 30*2^(2/3)*3^(1/6)*ArcTan[(3^(5 /6)*x)/(3^(1/3)*x + 2*2^(2/3)*(-1 + x^3)^(1/3))] + 40*2^(1/3)*Log[-x + 2^( 1/3)*(-1 + x^3)^(1/3)] - 10*6^(2/3)*Log[-3*x + 6^(2/3)*(-1 + x^3)^(1/3)] - 20*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3 )] + 5*6^(2/3)*Log[3*x^2 + 6^(2/3)*x*(-1 + x^3)^(1/3) + 2*6^(1/3)*(-1 + x^ 3)^(2/3)])/480
Time = 0.82 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7276, 2009}
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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6-8 x^3+8\right )}{x^6 \left (x^3-4\right ) \left (x^3-2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {\left (x^3-1\right )^{2/3}}{4 \left (x^3-4\right )}+\frac {\left (x^3-1\right )^{2/3}}{2 \left (x^3-2\right )}-\frac {\left (x^3-1\right )^{2/3}}{4 x^3}+\frac {\left (x^3-1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {\sqrt [3]{6} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{8 \sqrt [3]{2}}+\frac {\log \left (x^3-4\right )}{16 \sqrt [3]{6}}-\frac {\log \left (x^3-2\right )}{12\ 2^{2/3}}+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{4\ 2^{2/3}}-\frac {3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{2^{2/3}}-\sqrt [3]{x^3-1}\right )}{16 \sqrt [3]{2}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5}+\frac {\left (x^3-1\right )^{2/3}}{8 x^2}\) |
(-1 + x^3)^(2/3)/(8*x^2) + (-1 + x^3)^(5/3)/(5*x^5) - ArcTan[(1 + (2^(2/3) *x)/(-1 + x^3)^(1/3))/Sqrt[3]]/(2*2^(2/3)*Sqrt[3]) + (3^(1/6)*ArcTan[(1 + (6^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(8*2^(1/3)) + Log[-4 + x^3]/(16*6^ (1/3)) - Log[-2 + x^3]/(12*2^(2/3)) + Log[x/2^(1/3) - (-1 + x^3)^(1/3)]/(4 *2^(2/3)) - (3^(2/3)*Log[(3^(1/3)*x)/2^(2/3) - (-1 + x^3)^(1/3)])/(16*2^(1 /3))
3.28.74.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 6.99 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (13 x^{3}-8\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-5 x^{5} \left (\frac {2 \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-2 \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (2\right )\right ) 2^{\frac {1}{3}}}{3}+\arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} 2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{6}}+\frac {\left (\ln \left (\frac {-3^{\frac {1}{3}} 2^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {3^{\frac {2}{3}} 2^{\frac {2}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x 3^{\frac {1}{3}} 2^{\frac {1}{3}}+4 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) 6^{\frac {2}{3}}}{3}\right )}{80 x^{5}}\) | \(260\) |
1/80*(2*(13*x^3-8)*(x^3-1)^(2/3)-5*x^5*(2/3*(-2*arctan(1/3*3^(1/2)/x*(x+2* 2^(1/3)*(x^3-1)^(1/3)))*3^(1/2)-2*ln((-2^(2/3)*x+2*((-1+x)*(x^2+x+1))^(1/3 ))/x)+ln((2^(2/3)*x*((-1+x)*(x^2+x+1))^(1/3)+2^(1/3)*x^2+2*((-1+x)*(x^2+x+ 1))^(2/3))/x^2)+ln(2))*2^(1/3)+arctan(1/9*3^(1/2)*(2*3^(2/3)*2^(2/3)*(x^3- 1)^(1/3)+3*x)/x)*2^(2/3)*3^(1/6)+1/3*(ln((-3^(1/3)*2^(1/3)*x+2*((-1+x)*(x^ 2+x+1))^(1/3))/x)-1/2*ln((3^(2/3)*2^(2/3)*x^2+2*((-1+x)*(x^2+x+1))^(1/3)*x *3^(1/3)*2^(1/3)+4*((-1+x)*(x^2+x+1))^(2/3))/x^2))*6^(2/3)))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (195) = 390\).
Time = 3.82 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.08 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\frac {30 \cdot 6^{\frac {1}{6}} \sqrt {2} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {6^{\frac {1}{6}} {\left (24 \cdot 6^{\frac {2}{3}} \sqrt {2} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 22 \, x^{4} + 8 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 36 \, \sqrt {2} \left (-1\right )^{\frac {1}{3}} {\left (109 \, x^{8} - 116 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 6^{\frac {1}{3}} \sqrt {2} {\left (1189 \, x^{9} - 2064 \, x^{6} + 912 \, x^{3} - 64\right )}\right )}}{6 \, {\left (971 \, x^{9} - 960 \, x^{6} - 48 \, x^{3} + 64\right )}}\right ) + 10 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {18 \cdot 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 4\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 4}\right ) - 5 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {12 \cdot 6^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 6^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (109 \, x^{6} - 116 \, x^{3} + 16\right )} - 18 \, {\left (11 \, x^{5} - 8 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 8 \, x^{3} + 16}\right ) + 40 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 20 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (13 \, x^{3} - 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{1440 \, x^{5}} \]
1/1440*(30*6^(1/6)*sqrt(2)*(-1)^(1/3)*x^5*arctan(1/6*6^(1/6)*(24*6^(2/3)*s qrt(2)*(-1)^(2/3)*(5*x^7 - 22*x^4 + 8*x)*(x^3 - 1)^(2/3) - 36*sqrt(2)*(-1) ^(1/3)*(109*x^8 - 116*x^5 + 16*x^2)*(x^3 - 1)^(1/3) + 6^(1/3)*sqrt(2)*(118 9*x^9 - 2064*x^6 + 912*x^3 - 64))/(971*x^9 - 960*x^6 - 48*x^3 + 64)) + 10* 6^(2/3)*(-1)^(1/3)*x^5*log((18*6^(1/3)*(-1)^(2/3)*(x^3 - 1)^(1/3)*x^2 - 6^ (2/3)*(-1)^(1/3)*(x^3 - 4) - 36*(x^3 - 1)^(2/3)*x)/(x^3 - 4)) - 5*6^(2/3)* (-1)^(1/3)*x^5*log(-(12*6^(2/3)*(-1)^(1/3)*(5*x^4 - 2*x)*(x^3 - 1)^(2/3) - 6^(1/3)*(-1)^(2/3)*(109*x^6 - 116*x^3 + 16) - 18*(11*x^5 - 8*x^2)*(x^3 - 1)^(1/3))/(x^6 - 8*x^3 + 16)) + 40*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)* (12*4^(2/3)*sqrt(3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 4^(1/3)*sqrt(3 )*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*sqrt(3)*(19*x^8 - 22*x^5 + 4*x^2)*( x^3 - 1)^(1/3))/(53*x^9 - 48*x^6 - 12*x^3 + 8)) + 20*4^(2/3)*x^5*log((6*4^ (1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 - 1)^(2/3)*x)/(x^3 - 2)) - 10*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/ 3)*(19*x^6 - 22*x^3 + 4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) + 36*(13*x^3 - 8)*(x^3 - 1)^(2/3))/x^5
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - 8 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} - 8 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8-8 x^3+x^6\right )}{x^6 \left (-4+x^3\right ) \left (-2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6-8\,x^3+8\right )}{x^6\,\left (x^3-2\right )\,\left (x^3-4\right )} \,d x \]