Integrand size = 30, antiderivative size = 214 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (8+7 x^3\right )}{10 x^5}+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {2}{3} \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-\frac {1}{3} \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right ) \]
1/10*(x^3-1)^(2/3)*(7*x^3+8)/x^5+1/3*arctan(3^(1/2)*x/(x+2*(x^3-1)^(1/3))) *3^(1/2)-2/3*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/3)*(x^3-1)^(1/3)))*2^(1/3) -1/3*ln(-x+(x^3-1)^(1/3))+2/3*ln(-x+2^(1/3)*(x^3-1)^(1/3))*2^(1/3)+1/6*ln( x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))-1/3*ln(x^2+2^(1/3)*x*(x^3-1)^(1/3)+2^(2 /3)*(x^3-1)^(2/3))*2^(1/3)
Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.98 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {1}{30} \left (\frac {3 \left (-1+x^3\right )^{2/3} \left (8+7 x^3\right )}{x^5}+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )-20 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )-10 \log \left (-x+\sqrt [3]{-1+x^3}\right )+20 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )+5 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )-10 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )\right ) \]
((3*(-1 + x^3)^(2/3)*(8 + 7*x^3))/x^5 + 10*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3)^(1/3))] - 20*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/ 3)*(-1 + x^3)^(1/3))] - 10*Log[-x + (-1 + x^3)^(1/3)] + 20*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^3)^(1/3)] + 5*Log[x^2 + x*(-1 + x^3)^(1/3) + (-1 + x^3)^ (2/3)] - 10*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x ^3)^(2/3)])/30
Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6+2 x^3+8\right )}{x^6 \left (x^3-2\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {4 \left (x^3-1\right )^{2/3}}{x^3-2}-\frac {3 \left (x^3-1\right )^{2/3}}{x^3}-\frac {4 \left (x^3-1\right )^{2/3}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\sqrt {3} \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+\frac {4 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 \sqrt [3]{2} \arctan \left (\frac {\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (x^3-2\right )+\sqrt [3]{2} \log \left (\frac {x}{\sqrt [3]{2}}-\sqrt [3]{x^3-1}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {4 \left (x^3-1\right )^{5/3}}{5 x^5}+\frac {3 \left (x^3-1\right )^{2/3}}{2 x^2}\) |
(3*(-1 + x^3)^(2/3))/(2*x^2) - (4*(-1 + x^3)^(5/3))/(5*x^5) + (4*ArcTan[(1 + (2*x)/(-1 + x^3)^(1/3))/Sqrt[3]])/Sqrt[3] - Sqrt[3]*ArcTan[(1 + (2*x)/( -1 + x^3)^(1/3))/Sqrt[3]] - (2*2^(1/3)*ArcTan[(1 + (2^(2/3)*x)/(-1 + x^3)^ (1/3))/Sqrt[3]])/Sqrt[3] - (2^(1/3)*Log[-2 + x^3])/3 + 2^(1/3)*Log[x/2^(1/ 3) - (-1 + x^3)^(1/3)] - Log[-x + (-1 + x^3)^(1/3)]/2
3.26.47.3.1 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 = 2.38 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.04
| method | result | size |
| pseudoelliptic | \(\frac {20 \sqrt {3}\, 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{5}+20 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )-10 \,2^{\frac {1}{3}} x^{5} \ln \left (\frac {2^{\frac {2}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-10 \,2^{\frac {1}{3}} x^{5} \ln \left (2\right )-10 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+5 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}+21 x^{3} \left (x^{3}-1\right )^{\frac {2}{3}}+24 \left (x^{3}-1\right )^{\frac {2}{3}}}{30 x^{5}}\) | \(222\) |
1/30*(20*3^(1/2)*2^(1/3)*arctan(1/3*3^(1/2)/x*(x+2*2^(1/3)*(x^3-1)^(1/3))) *x^5-10*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^3-1)^(1/3)))*x^5+20*2^(1/3)*x ^5*ln((-2^(2/3)*x+2*(x^3-1)^(1/3))/x)-10*2^(1/3)*x^5*ln((2^(2/3)*x*(x^3-1) ^(1/3)+2^(1/3)*x^2+2*(x^3-1)^(2/3))/x^2)-10*2^(1/3)*x^5*ln(2)-10*ln((-x+(x ^3-1)^(1/3))/x)*x^5+5*ln((x^2+x*(x^3-1)^(1/3)+(x^3-1)^(2/3))/x^2)*x^5+21*x ^3*(x^3-1)^(2/3)+24*(x^3-1)^(2/3))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (165) = 330\).
Time = 14.28 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.68 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\frac {20 \, \sqrt {3} 2^{\frac {1}{3}} x^{5} \arctan \left (\frac {12 \, \sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )}}{3 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 30 \, \sqrt {3} x^{5} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) + 20 \cdot 2^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{3} - 2}\right ) - 10 \cdot 2^{\frac {1}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 15 \, x^{5} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 9 \, {\left (7 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]
1/90*(20*sqrt(3)*2^(1/3)*x^5*arctan(1/3*(12*sqrt(3)*2^(2/3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 6*sqrt(3)*2^(1/3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3) + sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8))/(53*x^9 - 48*x^6 - 12*x^3 + 8)) + 30*sqrt(3)*x^5*arctan(-(25382*sqrt(3)*(x^3 - 1)^(1/3)*x^2 - 13720*sqrt(3)*(x^3 - 1)^(2/3)*x + sqrt(3)*(5831*x^3 - 7200))/(58653*x^3 - 8000)) + 20*2^(1/3)*x^5*log(-(3*2^(2/3)*(x^3 - 1)^(1/3)*x^2 - 6*(x^3 - 1) ^(2/3)*x + 2^(1/3)*(x^3 - 2))/(x^3 - 2)) - 10*2^(1/3)*x^5*log((12*2^(1/3)* (2*x^4 - x)*(x^3 - 1)^(2/3) + 2^(2/3)*(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)) - 15*x^5*log(-3*(x^3 - 1)^(1/3)* x^2 + 3*(x^3 - 1)^(2/3)*x + 1) + 9*(7*x^3 + 8)*(x^3 - 1)^(2/3))/x^5
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} + 8\right )}{x^{6} \left (x^{3} - 2\right )}\, dx \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} + 8\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 2\right )} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (8+2 x^3+x^6\right )}{x^6 \left (-2+x^3\right )} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3+8\right )}{x^6\,\left (x^3-2\right )} \,d x \]