Integrand size = 25, antiderivative size = 79 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
-1/3*arctan(3^(1/2)*x/(-x+2*(x^6-1)^(1/3)))*3^(1/2)+1/3*ln(x+(x^6-1)^(1/3) )-1/6*ln(x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2/3))
Time = 1.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^6}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (x+\sqrt [3]{-1+x^6}\right )-\frac {1}{6} \log \left (x^2-x \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
-(ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^6)^(1/3))]/Sqrt[3]) + Log[x + (-1 + x ^6)^(1/3)]/3 - Log[x^2 - x*(-1 + x^6)^(1/3) + (-1 + x^6)^(2/3)]/6
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.
| \(\displaystyle \int \frac {x^7+x}{\left (x^6-1\right )^{2/3} \left (x^6+x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x \left (x^6+1\right )}{\left (x^6-1\right )^{2/3} \left (x^6+x^3-1\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {x}{\left (x^6-1\right )^{2/3}}+\frac {\left (2-x^3\right ) x}{\left (x^6-1\right )^{2/3} \left (x^6+x^3-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {x}{\left (-2 x^3+\sqrt {5}-1\right ) \left (x^6-1\right )^{2/3}}dx-\frac {4 \int \frac {x}{\left (-2 x^3+\sqrt {5}-1\right ) \left (x^6-1\right )^{2/3}}dx}{\sqrt {5}}-\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {x}{\left (2 x^3+\sqrt {5}+1\right ) \left (x^6-1\right )^{2/3}}dx-\frac {4 \int \frac {x}{\left (2 x^3+\sqrt {5}+1\right ) \left (x^6-1\right )^{2/3}}dx}{\sqrt {5}}+\frac {\left (1-x^6\right )^{2/3} x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^6\right )}{2 \left (x^6-1\right )^{2/3}}\) |
3.11.54.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 3.48 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\ln \left (\frac {x +\left (x^{6}-1\right )^{\frac {1}{3}}}{x}\right )}{3}-\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )}{3}\) | \(71\) |
| trager | \(\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}+x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x +9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-3 x \left (x^{6}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1}{x^{6}+x^{3}-1}\right )-\frac {\ln \left (\frac {-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right )}{3}-\ln \left (\frac {-6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{6}-x^{6}+18 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x -9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{2}+9 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+x^{3}+6 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+1}{x^{6}+x^{3}-1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(433\) |
1/3*ln((x+(x^6-1)^(1/3))/x)-1/6*ln((x^2-x*(x^6-1)^(1/3)+(x^6-1)^(2/3))/x^2 )-1/3*3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^6-1)^(1/3))/x)
Time = 1.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{6} - 1\right )}}{x^{6} - 8 \, x^{3} - 1}\right ) + \frac {1}{6} \, \log \left (\frac {x^{6} + x^{3} + 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x - 1}{x^{6} + x^{3} - 1}\right ) \]
-1/3*sqrt(3)*arctan((4*sqrt(3)*(x^6 - 1)^(1/3)*x^2 + 2*sqrt(3)*(x^6 - 1)^( 2/3)*x + sqrt(3)*(x^6 - 1))/(x^6 - 8*x^3 - 1)) + 1/6*log((x^6 + x^3 + 3*(x ^6 - 1)^(1/3)*x^2 + 3*(x^6 - 1)^(2/3)*x - 1)/(x^6 + x^3 - 1))
\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int \frac {x \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{\left (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + x^{3} - 1\right )}\, dx \]
Integral(x*(x**2 + 1)*(x**4 - x**2 + 1)/(((x - 1)*(x + 1)*(x**2 - x + 1)*( x**2 + x + 1))**(2/3)*(x**6 + x**3 - 1)), x)
\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int { \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}} \,d x } \]
\[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int { \frac {x^{7} + x}{{\left (x^{6} + x^{3} - 1\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}}} \,d x } \]
Timed out. \[ \int \frac {x+x^7}{\left (-1+x^6\right )^{2/3} \left (-1+x^3+x^6\right )} \, dx=\int \frac {x^7+x}{{\left (x^6-1\right )}^{2/3}\,\left (x^6+x^3-1\right )} \,d x \]