Integrand size = 27, antiderivative size = 81 \[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{-1+x^4}}\right )+\frac {1}{2} \log \left (-2 x+\sqrt [3]{-1+x^4}\right )-\frac {1}{4} \log \left (4 x^2+2 x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]
-1/2*3^(1/2)*arctan(3^(1/2)*x/(x+(x^4-1)^(1/3)))+1/2*ln(-2*x+(x^4-1)^(1/3) )-1/4*ln(4*x^2+2*x*(x^4-1)^(1/3)+(x^4-1)^(2/3))
Time = 1.18 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00 \[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=-\frac {1}{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+\sqrt [3]{-1+x^4}}\right )+\frac {1}{2} \log \left (-2 x+\sqrt [3]{-1+x^4}\right )-\frac {1}{4} \log \left (4 x^2+2 x \sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \]
-1/2*(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + (-1 + x^4)^(1/3))]) + Log[-2*x + (-1 + x^4)^(1/3)]/2 - Log[4*x^2 + 2*x*(-1 + x^4)^(1/3) + (-1 + x^4)^(2/3)]/4
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.
\(\displaystyle \int \frac {x^4+3}{\sqrt [3]{x^4-1} \left (x^4-8 x^3-1\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{x^4-1}}+\frac {4 \left (2 x^3+1\right )}{\sqrt [3]{x^4-1} \left (x^4-8 x^3-1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{\sqrt [3]{x^4-1} \left (x^4-8 x^3-1\right )}dx+8 \int \frac {x^3}{\sqrt [3]{x^4-1} \left (x^4-8 x^3-1\right )}dx+\frac {x \sqrt [3]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{3},\frac {5}{4},x^4\right )}{\sqrt [3]{x^4-1}}\) |
3.11.82.3.1 Defintions of rubi rules used
Time = 4.69 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {4 x^{2}+2 x \left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{4}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +\left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )}{2}+\frac {\ln \left (\frac {-2 x +\left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right )}{2}\) | \(73\) |
trager | \(\frac {\ln \left (-\frac {32 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x -8 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+32 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+x^{4}-4 \left (x^{4}-1\right )^{\frac {2}{3}} x +4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+8 x^{3}-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-1}{x^{4}-8 x^{3}-1}\right )}{2}+\operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (\frac {32 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}} x +16 \left (x^{4}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+16 x^{3} \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-x^{4}+2 \left (x^{4}-1\right )^{\frac {2}{3}} x +4 \left (x^{4}-1\right )^{\frac {1}{3}} x^{2}+2 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+1}{x^{4}-8 x^{3}-1}\right )\) | \(320\) |
-1/4*ln((4*x^2+2*x*(x^4-1)^(1/3)+(x^4-1)^(2/3))/x^2)+1/2*3^(1/2)*arctan(1/ 3*3^(1/2)/x*(x+(x^4-1)^(1/3)))+1/2*ln((-2*x+(x^4-1)^(1/3))/x)
Time = 1.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.38 \[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (-\frac {8 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{4} - 8 \, x^{3} - 1\right )}}{3 \, {\left (x^{4} + 8 \, x^{3} - 1\right )}}\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - 8 \, x^{3} + 12 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - 8 \, x^{3} - 1}\right ) \]
-1/2*sqrt(3)*arctan(-1/3*(8*sqrt(3)*(x^4 - 1)^(1/3)*x^2 - 4*sqrt(3)*(x^4 - 1)^(2/3)*x + sqrt(3)*(x^4 - 8*x^3 - 1))/(x^4 + 8*x^3 - 1)) + 1/4*log((x^4 - 8*x^3 + 12*(x^4 - 1)^(1/3)*x^2 - 6*(x^4 - 1)^(2/3)*x - 1)/(x^4 - 8*x^3 - 1))
Timed out. \[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=\int { \frac {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}} \,d x } \]
\[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=\int { \frac {x^{4} + 3}{{\left (x^{4} - 8 \, x^{3} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}} \,d x } \]
Timed out. \[ \int \frac {3+x^4}{\sqrt [3]{-1+x^4} \left (-1-8 x^3+x^4\right )} \, dx=\int -\frac {x^4+3}{{\left (x^4-1\right )}^{1/3}\,\left (-x^4+8\,x^3+1\right )} \,d x \]