Integrand size = 29, antiderivative size = 97 \[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^8-x^3+1)^(1/3)))-ln(x+(x^8-x^3+1)^(1/3) )+1/2*ln(x^2-x*(x^8-x^3+1)^(1/3)+(x^8-x^3+1)^(2/3))
Time = 1.50 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{1-x^3+x^8}}\right )-\log \left (x+\sqrt [3]{1-x^3+x^8}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1-x^3+x^8}+\left (1-x^3+x^8\right )^{2/3}\right ) \]
Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x - 2*(1 - x^3 + x^8)^(1/3))] - Log[x + (1 - x ^3 + x^8)^(1/3)] + Log[x^2 - x*(1 - x^3 + x^8)^(1/3) + (1 - x^3 + x^8)^(2/ 3)]/2
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.
| \(\displaystyle \int \frac {5 x^8-3}{\left (x^8+1\right ) \sqrt [3]{x^8-x^3+1}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {5}{\sqrt [3]{x^8-x^3+1}}-\frac {8}{\left (x^8+1\right ) \sqrt [3]{x^8-x^3+1}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 5 \int \frac {1}{\sqrt [3]{x^8-x^3+1}}dx-\sqrt [8]{-1} \int \frac {1}{\left (\sqrt [8]{-1}-x\right ) \sqrt [3]{x^8-x^3+1}}dx-(-1)^{3/8} \int \frac {1}{\left ((-1)^{3/8}-x\right ) \sqrt [3]{x^8-x^3+1}}dx+(-1)^{5/8} \int \frac {1}{\left (-x-(-1)^{5/8}\right ) \sqrt [3]{x^8-x^3+1}}dx+(-1)^{7/8} \int \frac {1}{\left (-x-(-1)^{7/8}\right ) \sqrt [3]{x^8-x^3+1}}dx-\sqrt [8]{-1} \int \frac {1}{\left (x+\sqrt [8]{-1}\right ) \sqrt [3]{x^8-x^3+1}}dx-(-1)^{3/8} \int \frac {1}{\left (x+(-1)^{3/8}\right ) \sqrt [3]{x^8-x^3+1}}dx+(-1)^{5/8} \int \frac {1}{\left (x-(-1)^{5/8}\right ) \sqrt [3]{x^8-x^3+1}}dx+(-1)^{7/8} \int \frac {1}{\left (x-(-1)^{7/8}\right ) \sqrt [3]{x^8-x^3+1}}dx\) |
3.14.53.3.1 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 = 4.35 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {x +\left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}-x \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -2 \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )\) | \(91\) |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {x^{8}+\left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-2 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x +2 \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}+1}{x^{8}+1}\right )-\ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{8}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{8}+x^{8}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (x^{8}-x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{8}-x^{3}+1\right )^{\frac {1}{3}} x^{2}-x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1}{x^{8}+1}\right )\) | \(446\) |
-ln((x+(x^8-x^3+1)^(1/3))/x)+1/2*ln((x^2-x*(x^8-x^3+1)^(1/3)+(x^8-x^3+1)^( 2/3))/x^2)-3^(1/2)*arctan(1/3*3^(1/2)*(x-2*(x^8-x^3+1)^(1/3))/x)
Time = 1.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25 \[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=-\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{8} - x^{3} + 1\right )}}{x^{8} - 9 \, x^{3} + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{8} - x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 1}\right ) \]
-sqrt(3)*arctan((4*sqrt(3)*(x^8 - x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^8 - x^ 3 + 1)^(2/3)*x + sqrt(3)*(x^8 - x^3 + 1))/(x^8 - 9*x^3 + 1)) - 1/2*log((x^ 8 + 3*(x^8 - x^3 + 1)^(1/3)*x^2 + 3*(x^8 - x^3 + 1)^(2/3)*x + 1)/(x^8 + 1) )
\[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=\int \frac {5 x^{8} - 3}{\left (x^{8} + 1\right ) \sqrt [3]{x^{8} - x^{3} + 1}}\, dx \]
\[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=\int { \frac {5 \, x^{8} - 3}{{\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{8} + 1\right )}} \,d x } \]
\[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=\int { \frac {5 \, x^{8} - 3}{{\left (x^{8} - x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{8} + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-3+5 x^8}{\left (1+x^8\right ) \sqrt [3]{1-x^3+x^8}} \, dx=\int \frac {5\,x^8-3}{\left (x^8+1\right )\,{\left (x^8-x^3+1\right )}^{1/3}} \,d x \]