Integrand size = 25, antiderivative size = 131 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\frac {\sqrt [3]{1+x^6}}{x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \]
(x^6+1)^(1/3)/x+1/3*2^(1/3)*arctan(3^(1/2)*x/(-x+2^(2/3)*(x^6+1)^(1/3)))*3 ^(1/2)-1/3*2^(1/3)*ln(2*x+2^(2/3)*(x^6+1)^(1/3))+1/6*ln(-2*x^2+2^(2/3)*x*( x^6+1)^(1/3)-2^(1/3)*(x^6+1)^(2/3))*2^(1/3)
Time = 0.70 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\frac {\sqrt [3]{1+x^6}}{x}+\frac {\sqrt [3]{2} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^6}-\sqrt [3]{2} \left (1+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \]
(1 + x^6)^(1/3)/x + (2^(1/3)*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 + x^6)^(1 /3))])/Sqrt[3] - (2^(1/3)*Log[2*x + 2^(2/3)*(1 + x^6)^(1/3)])/3 + Log[-2*x ^2 + 2^(2/3)*x*(1 + x^6)^(1/3) - 2^(1/3)*(1 + x^6)^(2/3)]/(3*2^(2/3))
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.
\(\displaystyle \int \frac {\left (x^3-1\right ) \sqrt [3]{x^6+1}}{x^2 \left (x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {2 \sqrt [3]{x^6+1}}{3 (x+1)}+\frac {2 \sqrt [3]{x^6+1} (x+1)}{3 \left (x^2-x+1\right )}-\frac {\sqrt [3]{x^6+1}}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2}{3} \int \frac {\sqrt [3]{x^6+1}}{x+1}dx+\frac {2}{3} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 x-i \sqrt {3}-1}dx+\frac {2}{3} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt [3]{x^6+1}}{2 x+i \sqrt {3}-1}dx+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{6},\frac {5}{6},-x^6\right )}{x}\) |
3.19.92.3.1 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 = 8.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {2 \,2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (x^{6}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -2 \,2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{6}+1\right )^{\frac {1}{3}}}{x}\right ) x +2^{\frac {1}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} x \left (x^{6}+1\right )^{\frac {1}{3}}+\left (x^{6}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{6}+1\right )^{\frac {1}{3}}}{6 x}\) | \(110\) |
1/6*(2*2^(1/3)*3^(1/2)*arctan(1/3*3^(1/2)*(x-2^(2/3)*(x^6+1)^(1/3))/x)*x-2 *2^(1/3)*ln((2^(1/3)*x+(x^6+1)^(1/3))/x)*x+2^(1/3)*ln((2^(2/3)*x^2-2^(1/3) *x*(x^6+1)^(1/3)+(x^6+1)^(2/3))/x^2)*x+6*(x^6+1)^(1/3))/x
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (100) = 200\).
Time = 21.21 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.50 \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\frac {2 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} \left (-2\right )^{\frac {2}{3}} {\left (x^{14} - 14 \, x^{11} + 6 \, x^{8} - 14 \, x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} {\left (x^{13} - 2 \, x^{10} - 6 \, x^{7} - 2 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 51 \, x^{12} - 52 \, x^{9} + 51 \, x^{6} - 30 \, x^{3} + 1\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 93 \, x^{12} + 20 \, x^{9} - 93 \, x^{6} + 6 \, x^{3} + 1\right )}}\right ) + 2 \, \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {6 \, \left (-2\right )^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} - \left (-2\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )} - 6 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x^{3} + 1}\right ) - \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {3 \, \left (-2\right )^{\frac {2}{3}} {\left (x^{7} - 4 \, x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + \left (-2\right )^{\frac {1}{3}} {\left (x^{12} - 14 \, x^{9} + 6 \, x^{6} - 14 \, x^{3} + 1\right )} - 12 \, {\left (x^{8} - x^{5} + x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) + 18 \, {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{18 \, x} \]
1/18*(2*sqrt(3)*(-2)^(1/3)*x*arctan(1/3*(6*sqrt(3)*(-2)^(2/3)*(x^14 - 14*x ^11 + 6*x^8 - 14*x^5 + x^2)*(x^6 + 1)^(1/3) + 6*sqrt(3)*(-2)^(1/3)*(x^13 - 2*x^10 - 6*x^7 - 2*x^4 + x)*(x^6 + 1)^(2/3) + sqrt(3)*(x^18 - 30*x^15 + 5 1*x^12 - 52*x^9 + 51*x^6 - 30*x^3 + 1))/(x^18 + 6*x^15 - 93*x^12 + 20*x^9 - 93*x^6 + 6*x^3 + 1)) + 2*(-2)^(1/3)*x*log(-(6*(-2)^(1/3)*(x^6 + 1)^(1/3) *x^2 - (-2)^(2/3)*(x^6 + 2*x^3 + 1) - 6*(x^6 + 1)^(2/3)*x)/(x^6 + 2*x^3 + 1)) - (-2)^(1/3)*x*log(-(3*(-2)^(2/3)*(x^7 - 4*x^4 + x)*(x^6 + 1)^(2/3) + (-2)^(1/3)*(x^12 - 14*x^9 + 6*x^6 - 14*x^3 + 1) - 12*(x^8 - x^5 + x^2)*(x^ 6 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1)) + 18*(x^6 + 1)^(1/3))/x
\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\int \frac {\sqrt [3]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}{x^{2} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Integral(((x**2 + 1)*(x**4 - x**2 + 1))**(1/3)*(x - 1)*(x**2 + x + 1)/(x** 2*(x + 1)*(x**2 - x + 1)), x)
\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{{\left (x^{3} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^6}}{x^2 \left (1+x^3\right )} \, dx=\int \frac {\left (x^3-1\right )\,{\left (x^6+1\right )}^{1/3}}{x^2\,\left (x^3+1\right )} \,d x \]