Integrand size = 29, antiderivative size = 93 \[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-x^2+x^6}}\right )-\log \left (x+\sqrt [3]{-x^2+x^6}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-x^2+x^6}+\left (-x^2+x^6\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^6-x^2)^(1/3)))-ln(x+(x^6-x^2)^(1/3))+1/ 2*ln(x^2-x*(x^6-x^2)^(1/3)+(x^6-x^2)^(2/3))
\[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=\int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx \]
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.
| \(\displaystyle \int \frac {3 x^4+1}{\left (x^4+x-1\right ) \sqrt [3]{x^6-x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^4-1} \int -\frac {3 x^4+1}{x^{2/3} \left (-x^4-x+1\right ) \sqrt [3]{x^4-1}}dx}{\sqrt [3]{x^6-x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^4-1} \int \frac {3 x^4+1}{x^{2/3} \left (-x^4-x+1\right ) \sqrt [3]{x^4-1}}dx}{\sqrt [3]{x^6-x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^4-1} \int \frac {3 x^4+1}{\left (-x^4-x+1\right ) \sqrt [3]{x^4-1}}d\sqrt [3]{x}}{\sqrt [3]{x^6-x^2}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^4-1} \int \left (\frac {4-3 x}{\left (-x^4-x+1\right ) \sqrt [3]{x^4-1}}-\frac {3}{\sqrt [3]{x^4-1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^6-x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^4-1} \left (-4 \int \frac {1}{\sqrt [3]{x^4-1} \left (x^4+x-1\right )}d\sqrt [3]{x}+3 \int \frac {x}{\sqrt [3]{x^4-1} \left (x^4+x-1\right )}d\sqrt [3]{x}-\frac {3 \sqrt [3]{x} \sqrt [3]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{12},\frac {1}{3},\frac {13}{12},x^4\right )}{\sqrt [3]{x^4-1}}\right )}{\sqrt [3]{x^6-x^2}}\) |
3.13.90.3.1 Defintions of rubi rules used
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 6.90 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {x +\left (x^{6}-x^{2}\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {x^{2}-x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+\left (x^{6}-x^{2}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(87\) |
| trager | \(-\ln \left (-\frac {-680618 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}+11776766 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}-23512440 x^{5}+5104635 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+10436076 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10374438 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x +680618 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -20604254 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-41621028 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-20872152 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}-11776766 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +20377448 x^{2}+23512440 x}{x \left (x^{4}+x -1\right )}\right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (\frac {-391874 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{5}-4413744 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{5}+8848034 x^{5}+2939055 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{2}+5218038 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {2}{3}}+10405257 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}-x^{2}\right )^{\frac {1}{3}} x +391874 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x -5888383 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+10374438 \left (x^{6}-x^{2}\right )^{\frac {2}{3}}-10436076 x \left (x^{6}-x^{2}\right )^{\frac {1}{3}}+4413744 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x +1361236 x^{2}-8848034 x}{x \left (x^{4}+x -1\right )}\right )}{2}\) | \(386\) |
-ln((x+(x^6-x^2)^(1/3))/x)+1/2*ln((x^2-x*(x^6-x^2)^(1/3)+(x^6-x^2)^(2/3))/ x^2)-3^(1/2)*arctan(1/3*(-2*(x^6-x^2)^(1/3)+x)*3^(1/2)/x)
Time = 1.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.33 \[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{5} + x^{2} - x\right )} + 2 \, \sqrt {3} {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{5} - x^{2} - x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{5} + x^{2} + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} x - x + 3 \, {\left (x^{6} - x^{2}\right )}^{\frac {2}{3}}}{x^{5} + x^{2} - x}\right ) \]
-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^6 - x^2)^(1/3)*x + sqrt(3)*(x^5 + x^2 - x) + 2*sqrt(3)*(x^6 - x^2)^(2/3))/(x^5 - x^2 - x)) - 1/2*log((x^5 + x^2 + 3*(x^6 - x^2)^(1/3)*x - x + 3*(x^6 - x^2)^(2/3))/(x^5 + x^2 - x))
Timed out. \[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=\text {Timed out} \]
\[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=\int { \frac {3 \, x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{4} + x - 1\right )}} \,d x } \]
\[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=\int { \frac {3 \, x^{4} + 1}{{\left (x^{6} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{4} + x - 1\right )}} \,d x } \]
Timed out. \[ \int \frac {1+3 x^4}{\left (-1+x+x^4\right ) \sqrt [3]{-x^2+x^6}} \, dx=\int \frac {3\,x^4+1}{{\left (x^6-x^2\right )}^{1/3}\,\left (x^4+x-1\right )} \,d x \]