Integrand size = 25, antiderivative size = 289 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{2\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{4\ 2^{2/3} \sqrt [3]{3}} \]
1/5*(-x^3+1)*(x^3-1)^(2/3)/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)+1/4*2^(1/3)*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2* 2^(1/3)*(x^3-1)^(1/3)))+1/12*ln(-x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)-1/12*2^( 1/3)*3^(2/3)*ln(-3*x+2^(1/3)*3^(2/3)*(x^3-1)^(1/3))-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)+1/24*ln(3*x^2+2^(1/3)*3^(2/3) *x*(x^3-1)^(1/3)+2^(2/3)*3^(1/3)*(x^3-1)^(2/3))*2^(1/3)*3^(2/3)
Time = 0.62 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {1}{120} \left (-\frac {24 \left (-1+x^3\right )^{5/3}}{x^5}-10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+30 \sqrt [3]{2} \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )+10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )-10 \sqrt [3]{2} 3^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )-5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )+5 \sqrt [3]{2} 3^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
((-24*(-1 + x^3)^(5/3))/x^5 - 10*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2 *2^(1/3)*(-1 + x^3)^(1/3))] + 30*2^(1/3)*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/ 3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))] + 10*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^ 3)^(1/3)] - 10*2^(1/3)*3^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3) ] - 5*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2 /3)] + 5*2^(1/3)*3^(2/3)*Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)])/120
Time = 0.54 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6+4\right )}{x^6 \left (x^6-4\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\left (x^3-1\right )^{2/3}}{2 \left (x^3-2\right )}-\frac {\left (x^3-1\right )^{2/3}}{2 \left (x^3+2\right )}-\frac {\left (x^3-1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^3-2\right )}{12\ 2^{2/3}}+\frac {\log \left (x^3+2\right )}{4\ 2^{2/3} \sqrt [3]{3}}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{\frac {3}{2}} x-\sqrt [3]{x^3-1}\right )+\frac {\log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )}{4\ 2^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{5 x^5}\) |
-1/5*(-1 + x^3)^(5/3)/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 + (2^(2/3)*3^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/(2*2^(2/3)) - Log[-2 + x^3]/(12*2^(2/3)) + Log[2 + x ^3]/(4*2^(2/3)*3^(1/3)) - ((3/2)^(2/3)*Log[(3/2)^(1/3)*x - (-1 + x^3)^(1/3 )])/4 + Log[x/2^(1/3) - (-1 + x^3)^(1/3)]/(4*2^(2/3))
3.29.38.3.1 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 = 4.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.89
| method | result | size |
| pseudoelliptic | \(\frac {\left (-24 x^{3}+24\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-5 \,2^{\frac {1}{3}} x^{5} \left (\left (2 \ln \left (\frac {-2^{\frac {2}{3}} 3^{\frac {1}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}} x +2^{\frac {1}{3}} 3^{\frac {2}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (2\right )\right ) 3^{\frac {2}{3}}+6 \arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) 3^{\frac {1}{6}}-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {2}{3}} x +2 {\left (\left (-1+x \right ) \left (x^{2}+x +1\right )\right )}^{\frac {1}{3}}}{x}\right )+\ln \left (2\right )\right )}{120 x^{5}}\) | \(258\) |
1/120*((-24*x^3+24)*(x^3-1)^(2/3)-5*2^(1/3)*x^5*((2*ln((-2^(2/3)*3^(1/3)*x +2*((-1+x)*(x^2+x+1))^(1/3))/x)-ln((2^(2/3)*3^(1/3)*((-1+x)*(x^2+x+1))^(1/ 3)*x+2^(1/3)*3^(2/3)*x^2+2*((-1+x)*(x^2+x+1))^(2/3))/x^2)-ln(2))*3^(2/3)+6 *arctan(1/9*3^(1/2)*(2*2^(1/3)*3^(2/3)*(x^3-1)^(1/3)+3*x)/x)*3^(1/6)-2*arc tan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3-1)^(1/3)))*3^(1/2)+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)-2*ln((-2^ (2/3)*x+2*((-1+x)*(x^2+x+1))^(1/3))/x)+ln(2)))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (202) = 404\).
Time = 3.66 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.85 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) + 20 \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 ) + 10 \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 ) - 5 \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 ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 144 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{720 \, x^{5}} \]
1/720*(10*12^(2/3)*(-1)^(1/3)*x^5*log(-(18*12^(1/3)*(-1)^(2/3)*(x^3 - 1)^( 1/3)*x^2 + 12^(2/3)*(-1)^(1/3)*(x^3 + 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2) ) - 5*12^(2/3)*(-1)^(1/3)*x^5*log(-(6*12^(2/3)*(-1)^(1/3)*(4*x^4 - x)*(x^3 - 1)^(2/3) - 12^(1/3)*(-1)^(2/3)*(55*x^6 - 50*x^3 + 4) - 18*(7*x^5 - 4*x^ 2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3 + 4)) + 20*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)) + 10*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)) - 5*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)) - 60*12^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*12^(1/6)*(12*12^(2/ 3)*(-1)^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)^(2/3) + 36*(-1)^(1/3)*(55*x^ 8 - 50*x^5 + 4*x^2)*(x^3 - 1)^(1/3) - 12^(1/3)*(377*x^9 - 600*x^6 + 204*x^ 3 - 8))/(487*x^9 - 480*x^6 + 12*x^3 + 8)) - 144*(x^3 - 1)^(5/3))/x^5
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 4\right )}{x^{6} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}\, dx \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+4\right )}{x^6\,\left (x^6-4\right )} \,d x \]