Integrand size = 30, antiderivative size = 151 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
1/5*(x^3-1)^(2/3)*(-22*x^6-15*x^3+2)/x^5/(x^3+1)+7/3*arctan(3^(1/2)*x/(x+2 ^(2/3)*(x^3-1)^(1/3)))*2^(2/3)*3^(1/2)-7/3*ln(-2*x+2^(2/3)*(x^3-1)^(1/3))* 2^(2/3)+7/6*ln(2*x^2+2^(2/3)*x*(x^3-1)^(1/3)+2^(1/3)*(x^3-1)^(2/3))*2^(2/3 )
Time = 0.45 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (2-15 x^3-22 x^6\right )}{5 x^5 \left (1+x^3\right )}+\frac {7\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {7}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x^3}\right )+\frac {7 \log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x^3}+\sqrt [3]{2} \left (-1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
((-1 + x^3)^(2/3)*(2 - 15*x^3 - 22*x^6))/(5*x^5*(1 + x^3)) + (7*2^(2/3)*Ar cTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + x^3)^(1/3))])/Sqrt[3] - (7*2^(2/3)*Log [-2*x + 2^(2/3)*(-1 + x^3)^(1/3)])/3 + (7*Log[2*x^2 + 2^(2/3)*x*(-1 + x^3) ^(1/3) + 2^(1/3)*(-1 + x^3)^(2/3)])/(3*2^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(468\) vs. \(2(151)=302\).
Time = 2.03 (sec) , antiderivative size = 468, normalized size of antiderivative = 3.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7293, 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.
| \(\displaystyle \int \frac {\left (x^3-1\right )^{2/3} \left (x^6+2 x^3-2\right )}{x^6 \left (x^3+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (x^3-1\right )^{2/3}}{3 (x+1)}-\frac {\left (x^3-1\right )^{2/3}}{3 (x+1)^2}+\frac {6 \left (x^3-1\right )^{2/3}}{x^3}-\frac {2 \left (x^3-1\right )^{2/3}}{x^6}+\frac {\left (x^3-1\right )^{2/3} (x-1)}{\left (x^2-x+1\right )^2}+\frac {(8 x-15) \left (x^3-1\right )^{2/3}}{3 \left (x^2-x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {8\ 2^{2/3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {8\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {13\ 2^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {8\ 2^{2/3} \arctan \left (\frac {1-2^{2/3} \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {x \left (x^3-1\right )^{2/3}}{x^3+1}-\frac {5}{9} 2^{2/3} \log \left (x^3+1\right )+\frac {5 \log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {8}{9} 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}\right )-\frac {4}{9} 2^{2/3} \log \left (\frac {2^{2/3} (1-x)^2}{\left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{x^3-1}}+1\right )+\frac {2}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )-\frac {43 \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3-1}\right )}{9 \sqrt [3]{2}}+\frac {4}{3} 2^{2/3} \log \left (\sqrt [3]{x^3-1}+\sqrt [3]{2}\right )-\frac {4}{3} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-1}-x+1\right )-\frac {2 \left (x^3-1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3-1\right )^{2/3}}{x^2}+\frac {4}{9} 2^{2/3} \log \left ((1-x) (x+1)^2\right )\) |
(-3*(-1 + x^3)^(2/3))/x^2 - (2*(-1 + x^3)^(5/3))/(5*x^5) - (x*(-1 + x^3)^( 2/3))/(1 + x^3) + (8*2^(2/3)*ArcTan[(1 - (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3 ))/Sqrt[3]])/(3*Sqrt[3]) - (8*2^(2/3)*ArcTan[(1 + (2*2^(1/3)*(1 - x))/(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (13*2^(2/3)*ArcTan[(1 + (2*2^(1/3)*x )/(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (8*2^(2/3)*ArcTan[(1 - 2^(2/3) *(-1 + x^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]) + (4*2^(2/3)*Log[(1 - x)*(1 + x)^ 2])/9 + (5*Log[1 + x^3])/(3*2^(1/3)) - (5*2^(2/3)*Log[1 + x^3])/9 + (8*2^( 2/3)*Log[1 - (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3)])/9 - (4*2^(2/3)*Log[1 + ( 2^(2/3)*(1 - x)^2)/(-1 + x^3)^(2/3) + (2^(1/3)*(1 - x))/(-1 + x^3)^(1/3)]) /9 - (43*Log[2^(1/3)*x - (-1 + x^3)^(1/3)])/(9*2^(1/3)) + (2*2^(2/3)*Log[2 ^(1/3)*x - (-1 + x^3)^(1/3)])/9 + (4*2^(2/3)*Log[2^(1/3) + (-1 + x^3)^(1/3 )])/3 - (4*2^(2/3)*Log[1 - x + 2^(2/3)*(-1 + x^3)^(1/3)])/3
3.21.91.3.1 Defintions of rubi rules used
Time = 13.60 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {35 x^{5} \left (x^{3}+1\right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {2}{3}}-6 \left (x^{3}-1\right )^{\frac {2}{3}} \left (22 x^{6}+15 x^{3}-2\right )}{30 x^{8}+30 x^{5}}\) | \(131\) |
| risch | \(\text {Expression too large to display}\) | \(836\) |
| trager | \(\text {Expression too large to display}\) | \(1151\) |
(35*x^5*(x^3+1)*(-2*arctan(1/3*3^(1/2)/x*(x+2^(2/3)*(x^3-1)^(1/3)))*3^(1/2 )+ln((2^(2/3)*x^2+2^(1/3)*x*(x^3-1)^(1/3)+(x^3-1)^(2/3))/x^2)-2*ln((-2^(1/ 3)*x+(x^3-1)^(1/3))/x))*2^(2/3)-6*(x^3-1)^(2/3)*(22*x^6+15*x^3-2))/(30*x^8 +30*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (117) = 234\).
Time = 1.94 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.95 \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=-\frac {70 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 70 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) + 35 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} + x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 18 \, {\left (22 \, x^{6} + 15 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} + x^{5}\right )}} \]
-1/90*(70*sqrt(3)*(-4)^(1/3)*(x^8 + x^5)*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)* (5*x^7 + 4*x^4 - x)*(x^3 - 1)^(2/3) + 6*sqrt(3)*(-4)^(1/3)*(19*x^8 - 16*x^ 5 + x^2)*(x^3 - 1)^(1/3) - sqrt(3)*(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x ^9 - 105*x^6 + 3*x^3 + 1)) - 70*(-4)^(1/3)*(x^8 + x^5)*log(-(3*(-4)^(2/3)* (x^3 - 1)^(1/3)*x^2 - 6*(x^3 - 1)^(2/3)*x + (-4)^(1/3)*(x^3 + 1))/(x^3 + 1 )) + 35*(-4)^(1/3)*(x^8 + x^5)*log(-(6*(-4)^(1/3)*(5*x^4 - x)*(x^3 - 1)^(2 /3) - (-4)^(2/3)*(19*x^6 - 16*x^3 + 1) - 24*(2*x^5 - x^2)*(x^3 - 1)^(1/3)) /(x^6 + 2*x^3 + 1)) + 18*(22*x^6 + 15*x^3 - 2)*(x^3 - 1)^(2/3))/(x^8 + x^5 )
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2 x^{3} - 2\right )}{x^{6} \left (x + 1\right )^{2} \left (x^{2} - x + 1\right )^{2}}\, dx \]
Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + 2*x**3 - 2)/(x**6*(x + 1) **2*(x**2 - x + 1)**2), x)
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2 \, x^{3} - 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{3} + 1\right )}^{2} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right )^{2/3} \left (-2+2 x^3+x^6\right )}{x^6 \left (1+x^3\right )^2} \, dx=\int \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^6+2\,x^3-2\right )}{x^6\,{\left (x^3+1\right )}^2} \,d x \]