Integrand size = 27, antiderivative size = 74 \[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
-3^(1/2)*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 = 1.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^4}}\right )+\log \left (-x+\sqrt [3]{1+x^4}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}\right ) \]
-(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.
\(\displaystyle \int \frac {x^4-3}{\sqrt [3]{x^4+1} \left (x^4-x^3+1\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{x^4+1}}-\frac {4-x^3}{\sqrt [3]{x^4+1} \left (x^4-x^3+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {1}{\sqrt [3]{x^4+1} \left (x^4-x^3+1\right )}dx+\int \frac {x^3}{\sqrt [3]{x^4+1} \left (x^4-x^3+1\right )}dx+x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {5}{4},-x^4\right )\) |
3.10.76.3.1 Defintions of rubi rules used
Time = 4.52 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\ln \left (\frac {-x +\left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{4}+1\right )^{\frac {1}{3}}+\left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right )\) | \(69\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+2 \left (x^{4}+1\right )^{\frac {2}{3}} x +2 x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}+x^{3}+1}{x^{4}-x^{3}+1}\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{4}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-x^{4}-\left (x^{4}+1\right )^{\frac {2}{3}} x -x^{2} \left (x^{4}+1\right )^{\frac {1}{3}}-1}{x^{4}-x^{3}+1}\right )\) | \(287\) |
ln((-x+(x^4+1)^(1/3))/x)-1/2*ln((x^2+x*(x^4+1)^(1/3)+(x^4+1)^(2/3))/x^2)+3 ^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^4+1)^(1/3)))
Time = 1.03 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.49 \[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=-\sqrt {3} \arctan \left (-\frac {2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} - x^{3} + 1\right )}}{3 \, {\left (x^{4} + x^{3} + 1\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{4} - x^{3} + 3 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x + 1}{x^{4} - x^{3} + 1}\right ) \]
-sqrt(3)*arctan(-1/3*(2*sqrt(3)*(x^4 + 1)^(1/3)*x^2 - 2*sqrt(3)*(x^4 + 1)^ (2/3)*x + sqrt(3)*(x^4 - x^3 + 1))/(x^4 + x^3 + 1)) + 1/2*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))
Timed out. \[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=\int { \frac {x^{4} - 3}{{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=\int { \frac {x^{4} - 3}{{\left (x^{4} - x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {-3+x^4}{\sqrt [3]{1+x^4} \left (1-x^3+x^4\right )} \, dx=\int \frac {x^4-3}{{\left (x^4+1\right )}^{1/3}\,\left (x^4-x^3+1\right )} \,d x \]