Integrand size = 25, antiderivative size = 151 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+10 x^3-17 x^6\right )}{5 x^5 \left (-1+x^3\right )}+\frac {5\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5 \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
1/5*(x^3+1)^(2/3)*(-17*x^6+10*x^3+2)/x^5/(x^3-1)+5/3*2^(2/3)*arctan(3^(1/2 )*x/(x+2^(2/3)*(x^3+1)^(1/3)))*3^(1/2)-5/3*2^(2/3)*ln(-2*x+2^(2/3)*(x^3+1) ^(1/3))+5/6*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 = 0.44 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\frac {\left (1+x^3\right )^{2/3} \left (2+10 x^3-17 x^6\right )}{5 x^5 \left (-1+x^3\right )}+\frac {5\ 2^{2/3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {5}{3} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5 \log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{3 \sqrt [3]{2}} \]
((1 + x^3)^(2/3)*(2 + 10*x^3 - 17*x^6))/(5*x^5*(-1 + x^3)) + (5*2^(2/3)*Ar cTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))])/Sqrt[3] - (5*2^(2/3)*Log[ -2*x + 2^(2/3)*(1 + x^3)^(1/3)])/3 + (5*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. \(545\) vs. \(2(151)=302\).
Time = 2.16 (sec) , antiderivative size = 545, normalized size of antiderivative = 3.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {\left (x^3+1\right )^{2/3} \left (x^6+2\right )}{x^6 \left (x^3-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 \left (x^3+1\right )^{2/3}}{x-1}+\frac {\left (x^3+1\right )^{2/3}}{3 (x-1)^2}+\frac {4 \left (x^3+1\right )^{2/3}}{x^3}+\frac {2 \left (x^3+1\right )^{2/3}}{x^6}+\frac {\left (x^3+1\right )^{2/3} (x+1)}{\left (x^2+x+1\right )^2}+\frac {(6 x+11) \left (x^3+1\right )^{2/3}}{3 \left (x^2+x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2^{2/3} \sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2\ 2^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{\sqrt {3}}+2^{2/3} \sqrt {3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2^{2/3} \arctan \left (\frac {\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2\ 2^{2/3} \arctan \left (\frac {2^{2/3} \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {x \left (x^3+1\right )^{2/3}}{1-x^3}+\frac {2}{9} 2^{2/3} \log \left (1-x^3\right )-\frac {2}{27} 2^{2/3} \log \left (x^3-1\right )+\frac {\log \left (x^3-1\right )}{27 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {2^{2/3} (x+1)^2}{\left (x^3+1\right )^{2/3}}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )+\frac {2}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{x^3+1}}+1\right )+2^{2/3} \log \left (\sqrt [3]{2}-\sqrt [3]{x^3+1}\right )+\frac {2}{9} 2^{2/3} \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )-\frac {31 \log \left (\sqrt [3]{2} x-\sqrt [3]{x^3+1}\right )}{9 \sqrt [3]{2}}+\frac {\log \left (-2^{2/3} \sqrt [3]{x^3+1}+x+1\right )}{\sqrt [3]{2}}-\frac {3 \log \left (2^{2/3} \sqrt [3]{x^3+1}-x-1\right )}{\sqrt [3]{2}}-\frac {2 \left (x^3+1\right )^{5/3}}{5 x^5}-\frac {2 \left (x^3+1\right )^{2/3}}{x^2}+\frac {\log \left (-(1-x)^2 (x+1)\right )}{\sqrt [3]{2}}-\frac {\log \left ((1-x)^2 (x+1)\right )}{3 \sqrt [3]{2}}\) |
(-2*(1 + x^3)^(2/3))/x^2 + (x*(1 + x^3)^(2/3))/(1 - x^3) - (2*(1 + x^3)^(5 /3))/(5*x^5) + 2^(2/3)*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(1 + x^3)^(1/3))/ Sqrt[3]] - (2*2^(2/3)*ArcTan[(1 - (2*2^(1/3)*(1 + x))/(1 + x^3)^(1/3))/Sqr t[3]])/Sqrt[3] - (2^(2/3)*ArcTan[(1 + (2^(1/3)*(1 + x))/(1 + x^3)^(1/3))/S qrt[3]])/Sqrt[3] + 2^(2/3)*Sqrt[3]*ArcTan[(1 + (2^(1/3)*(1 + x))/(1 + x^3) ^(1/3))/Sqrt[3]] + (2*2^(2/3)*ArcTan[(1 + 2^(2/3)*(1 + x^3)^(1/3))/Sqrt[3] ])/Sqrt[3] + Log[-((1 - x)^2*(1 + x))]/2^(1/3) - Log[(1 - x)^2*(1 + x)]/(3 *2^(1/3)) + (2*2^(2/3)*Log[1 - x^3])/9 + Log[-1 + x^3]/(27*2^(1/3)) - (2*2 ^(2/3)*Log[-1 + x^3])/27 - (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)])/3 + (2*2^(2/3)*Log[1 + (2^(1/3 )*(1 + x))/(1 + x^3)^(1/3)])/3 + 2^(2/3)*Log[2^(1/3) - (1 + x^3)^(1/3)] - (31*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 + Log[1 + x - 2^(2/3)*(1 + x^3)^(1/3)]/2^(1/3) - (3*Log[-1 - x + 2^(2/3)*(1 + x^3)^(1/3)])/2^(1/3)
3.21.90.3.1 Defintions of rubi rules used
Time = 14.03 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87
| method | result | size |
| pseudoelliptic | \(\frac {25 x^{5} \left (x^{3}-1\right ) \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} x \left (x^{3}+1\right )^{\frac {1}{3}}+\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right )\right ) 2^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}} \left (17 x^{6}-10 x^{3}-2\right )}{30 x^{8}-30 x^{5}}\) | \(131\) |
| risch | \(\text {Expression too large to display}\) | \(938\) |
| trager | \(\text {Expression too large to display}\) | \(1127\) |
(25*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)*(17*x^6-10*x^3-2))/(30*x^8 -30*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (117) = 234\).
Time = 1.89 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.97 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=-\frac {50 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 50 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 25 \, \left (-4\right )^{\frac {1}{3}} {\left (x^{8} - x^{5}\right )} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 18 \, {\left (17 \, x^{6} - 10 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, {\left (x^{8} - x^{5}\right )}} \]
-1/90*(50*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)) - 50*(-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) ) + 25*(-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*(17*x^6 - 10*x^3 - 2)*(x^3 + 1)^(2/3))/(x^8 - x^5)
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 2\right )}{x^{6} \left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
\[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int { \frac {{\left (x^{6} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{3} - 1\right )}^{2} x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^6\right )}{x^6 \left (-1+x^3\right )^2} \, dx=\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+2\right )}{x^6\,{\left (x^3-1\right )}^2} \,d x \]