Integrand size = 30, antiderivative size = 112 \[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^3-x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3-x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^3-x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^3-x^4}+\left (-1+x^3-x^4\right )^{2/3}\right ) \]
3*(-x^4+x^3-1)^(1/3)/x+3^(1/2)*arctan(3^(1/2)*x/(x+2*(-x^4+x^3-1)^(1/3)))+ ln(-x+(-x^4+x^3-1)^(1/3))-1/2*ln(x^2+x*(-x^4+x^3-1)^(1/3)+(-x^4+x^3-1)^(2/ 3))
Time = 0.45 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\frac {3 \sqrt [3]{-1+x^3-x^4}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3-x^4}}\right )+\log \left (-x+\sqrt [3]{-1+x^3-x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^3-x^4}+\left (-1+x^3-x^4\right )^{2/3}\right ) \]
(3*(-1 + x^3 - x^4)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^3 - x^4)^(1/3))] + Log[-x + (-1 + x^3 - x^4)^(1/3)] - Log[x^2 + x*(-1 + x^3 - x^4)^(1/3) + (-1 + x^3 - x^4)^(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.
| \(\displaystyle \int \frac {\sqrt [3]{-x^4+x^3-1} \left (x^4-3\right )}{x^2 \left (x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {4 x^2 \sqrt [3]{-x^4+x^3-1}}{x^4+1}-\frac {3 \sqrt [3]{-x^4+x^3-1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (-1)^{3/4} \int \frac {\sqrt [3]{-x^4+x^3-1}}{\sqrt [4]{-1}-x}dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-x^4+x^3-1}}{-x-(-1)^{3/4}}dx+(-1)^{3/4} \int \frac {\sqrt [3]{-x^4+x^3-1}}{x+\sqrt [4]{-1}}dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-x^4+x^3-1}}{x-(-1)^{3/4}}dx-3 \int \frac {\sqrt [3]{-x^4+x^3-1}}{x^2}dx\) |
3.17.54.3.1 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 = 3.89 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x -\ln \left (\frac {x^{2}+x \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}+\left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +2 \ln \left (\frac {-x +\left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x +6 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}}{2 x}\) | \(114\) |
| trager | \(\frac {3 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}}}{x}+\ln \left (-\frac {-135 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+270 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+141 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +6 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-57 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-2 \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x +49 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} x^{2}-17 x^{3}-135 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+51 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{x^{4}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {-288 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+576 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-147 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+141 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x -147 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}+198 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-17 x^{4}+49 \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}} x -2 \left (-x^{4}+x^{3}-1\right )^{\frac {1}{3}} x^{2}+17 x^{3}-288 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-147 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-17}{x^{4}+1}\right )\) | \(433\) |
| risch | \(-\frac {3 \left (x^{4}-x^{3}+1\right )}{x \left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}}}+\frac {\left (\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}+x^{8}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-2 x^{7}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}+x^{6}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 x^{4}-2 x^{3}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (x^{4}+1\right ) \left (x^{4}-x^{3}+1\right )}\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 )^{2} x^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-x^{8}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 x^{7}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{5}-2 x^{6}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {2}{3}} x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 x^{4}+3 x^{3}+3 \left (x^{8}-2 x^{7}+x^{6}+2 x^{4}-2 x^{3}+1\right )^{\frac {1}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (x^{4}+1\right ) \left (x^{4}-x^{3}+1\right )}\right )\right ) {\left (\left (x^{4}-x^{3}+1\right )^{2}\right )}^{\frac {1}{3}}}{\left (-x^{4}+x^{3}-1\right )^{\frac {2}{3}}}\) | \(688\) |
1/2*(-2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(-x^4+x^3-1)^(1/3)))*x-ln((x^2+x *(-x^4+x^3-1)^(1/3)+(-x^4+x^3-1)^(2/3))/x^2)*x+2*ln((-x+(-x^4+x^3-1)^(1/3) )/x)*x+6*(-x^4+x^3-1)^(1/3))/x
Time = 1.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=-\frac {2 \, \sqrt {3} x \arctan \left (\frac {\sqrt {3} x^{3} - 2 \, \sqrt {3} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4 \, \sqrt {3} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {2}{3}} x}{8 \, x^{4} - 9 \, x^{3} + 8}\right ) - x \log \left (\frac {x^{4} - 3 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {2}{3}} x + 1}{x^{4} + 1}\right ) - 6 \, {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]
-1/2*(2*sqrt(3)*x*arctan((sqrt(3)*x^3 - 2*sqrt(3)*(-x^4 + x^3 - 1)^(1/3)*x ^2 + 4*sqrt(3)*(-x^4 + x^3 - 1)^(2/3)*x)/(8*x^4 - 9*x^3 + 8)) - x*log((x^4 - 3*(-x^4 + x^3 - 1)^(1/3)*x^2 + 3*(-x^4 + x^3 - 1)^(2/3)*x + 1)/(x^4 + 1 )) - 6*(-x^4 + x^3 - 1)^(1/3))/x
\[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\int \frac {\left (x^{4} - 3\right ) \sqrt [3]{- x^{4} + x^{3} - 1}}{x^{2} \left (x^{4} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3\right )} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 3\right )} {\left (-x^{4} + x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{-1+x^3-x^4} \left (-3+x^4\right )}{x^2 \left (1+x^4\right )} \, dx=\int \frac {\left (x^4-3\right )\,{\left (-x^4+x^3-1\right )}^{1/3}}{x^2\,\left (x^4+1\right )} \,d x \]