Integrand size = 30, antiderivative size = 90 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
3/2*(x^4+1)^(2/3)/x^2-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^4+1)^(1/3)))+ln(-x+ (x^4+1)^(1/3))-1/2*ln(x^2+x*(x^4+1)^(1/3)+(x^4+1)^(2/3))
Time = 1.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
(3*(1 + x^4)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^4)^ (1/3))] + Log[-x + (1 + x^4)^(1/3)] - Log[x^2 + x*(1 + x^4)^(1/3) + (1 + 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.
| \(\displaystyle \int \frac {\left (x^4-3\right ) \left (x^4+1\right )^{2/3}}{x^3 \left (x^4-x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(4 x-3) \left (x^4+1\right )^{2/3}}{x^4-x^3+1}-\frac {3 \left (x^4+1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {\left (x^4+1\right )^{2/3}}{x^4-x^3+1}dx+4 \int \frac {x \left (x^4+1\right )^{2/3}}{x^4-x^3+1}dx+\frac {2 \sqrt {2} 3^{3/4} \left (1-\sqrt [3]{x^4+1}\right ) \sqrt {\frac {\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{x^4+1}+\sqrt {3}+1}{-\sqrt [3]{x^4+1}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-\sqrt [3]{x^4+1}}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}} x^2}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{x^4+1}\right ) \sqrt {\frac {\left (x^4+1\right )^{2/3}+\sqrt [3]{x^4+1}+1}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{x^4+1}+\sqrt {3}+1}{-\sqrt [3]{x^4+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-\sqrt [3]{x^4+1}}{\left (-\sqrt [3]{x^4+1}-\sqrt {3}+1\right )^2}} x^2}+\frac {6 x^2}{-\sqrt [3]{x^4+1}-\sqrt {3}+1}+\frac {3 \left (x^4+1\right )^{2/3}}{2 x^2}\) |
3.13.34.3.1 Defintions of rubi rules used
Time = 6.17 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(95\) |
| risch | \(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x +2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+1}{x^{4}-x^{3}+1}\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right )\) | \(299\) |
| trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}}}{2 x^{2}}+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x -9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-2067 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+2746 x^{4}+8010 \left (x^{4}+1\right )^{\frac {2}{3}} x +8010 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+4119 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-10989 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+2746}{x^{4}-x^{3}+1}\right )-3 \ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-11673 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-\ln \left (\frac {10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-20610 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}+9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {2}{3}} x +9621 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}+1\right )^{\frac {1}{3}} x^{2}-11673 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+7554 x^{4}+11217 \left (x^{4}+1\right )^{\frac {2}{3}} x +11217 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+2518 x^{3}+10305 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+17859 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+7554}{x^{4}-x^{3}+1}\right )\) | \(606\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^4+1)^(1/3)))*x^2+2*ln((-x+(x^4 +1)^(1/3))/x)*x^2-ln((x^2+x*(x^4+1)^(1/3)+(x^4+1)^(2/3))/x^2)*x^2+3*(x^4+1 )^(2/3))/x^2
Time = 1.85 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {13034 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 686 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (37 \, x^{4} + 6137 \, x^{3} + 37\right )}}{3 \, {\left (x^{4} + 6859 \, x^{3} + 1\right )}}\right ) - x^{2} \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
-1/2*(2*sqrt(3)*x^2*arctan(-1/3*(13034*sqrt(3)*(x^4 + 1)^(1/3)*x^2 - 686*s qrt(3)*(x^4 + 1)^(2/3)*x + sqrt(3)*(37*x^4 + 6137*x^3 + 37))/(x^4 + 6859*x ^3 + 1)) - x^2*log((x^4 - x^3 + 3*(x^4 + 1)^(1/3)*x^2 - 3*(x^4 + 1)^(2/3)* x + 1)/(x^4 - x^3 + 1)) - 3*(x^4 + 1)^(2/3))/x^2
Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (x^{4} - x^{3} + 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (1-x^3+x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{2/3}\,\left (x^4-3\right )}{x^3\,\left (x^4-x^3+1\right )} \,d x \]