Integrand size = 25, antiderivative size = 85 \[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^4}}\right )-\log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right ) \]
-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^4+x^2)^(1/3)))-ln(x+(x^4+x^2)^(1/3))+1/ 2*ln(x^2-x*(x^4+x^2)^(1/3)+(x^4+x^2)^(2/3))
Time = 1.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\frac {x^{2/3} \sqrt [3]{1+x^2} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}-2 \sqrt [3]{1+x^2}}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [3]{1+x^2}\right )+\log \left (x^{2/3}-\sqrt [3]{x} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )}{2 \sqrt [3]{x^2+x^4}} \]
(x^(2/3)*(1 + x^2)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) - 2* (1 + x^2)^(1/3))] - 2*Log[x^(1/3) + (1 + x^2)^(1/3)] + Log[x^(2/3) - x^(1/ 3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]))/(2*(x^2 + x^4)^(1/3))
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.89 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2467, 25, 2035, 7279, 2009}
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.
| \(\displaystyle \int \frac {x^2-1}{\left (x^2+x+1\right ) \sqrt [3]{x^4+x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{x^2+1} \int -\frac {1-x^2}{x^{2/3} \sqrt [3]{x^2+1} \left (x^2+x+1\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{x^2+1} \int \frac {1-x^2}{x^{2/3} \sqrt [3]{x^2+1} \left (x^2+x+1\right )}dx}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \frac {1-x^2}{\sqrt [3]{x^2+1} \left (x^2+x+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \int \left (\frac {x+2}{\sqrt [3]{x^2+1} \left (x^2+x+1\right )}-\frac {1}{\sqrt [3]{x^2+1}}\right )d\sqrt [3]{x}}{\sqrt [3]{x^4+x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{x^2+1} \left (\sqrt [3]{x} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},-\frac {2 x^2}{1-i \sqrt {3}},-x^2\right )+\sqrt [3]{x} \operatorname {AppellF1}\left (\frac {1}{6},1,\frac {1}{3},\frac {7}{6},-\frac {2 x^2}{1+i \sqrt {3}},-x^2\right )+\frac {\left (-\sqrt {3}+i\right ) x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {2 x^2}{1-i \sqrt {3}}\right )}{4 \left (\sqrt {3}+i\right )}+\frac {\left (\sqrt {3}+i\right ) x^{4/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x^2,-\frac {2 x^2}{1+i \sqrt {3}}\right )}{4 \left (-\sqrt {3}+i\right )}-\sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{3},\frac {7}{6},-x^2\right )\right )}{\sqrt [3]{x^4+x^2}}\) |
(-3*x^(2/3)*(1 + x^2)^(1/3)*(x^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/( 1 - I*Sqrt[3]), -x^2] + x^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, (-2*x^2)/(1 + I *Sqrt[3]), -x^2] + ((I - Sqrt[3])*x^(4/3)*AppellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 - I*Sqrt[3])])/(4*(I + Sqrt[3])) + ((I + Sqrt[3])*x^(4/3)*App ellF1[2/3, 1/3, 1, 5/3, -x^2, (-2*x^2)/(1 + I*Sqrt[3])])/(4*(I - Sqrt[3])) - x^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -x^2]))/(x^2 + x^4)^(1/3)
3.12.39.3.1 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]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(-\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x}{x}\right )+\frac {\ln \left (\frac {\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}-\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\left (-2 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )\) | \(87\) |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 x^{3}-3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}+3 x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}+2 x}{x \left (x^{2}+x +1\right )}\right )-\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}-x}{x \left (x^{2}+x +1\right )}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {2}{3}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x -x^{2}-x}{x \left (x^{2}+x +1\right )}\right )\) | \(371\) |
-ln(((x^2*(x^2+1))^(1/3)+x)/x)+1/2*ln(((x^2*(x^2+1))^(2/3)-(x^2*(x^2+1))^( 1/3)*x+x^2)/x^2)-3^(1/2)*arctan(1/3*(-2*(x^2*(x^2+1))^(1/3)+x)*3^(1/2)/x)
Time = 0.78 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.25 \[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{8 \, x^{3} - x^{2} + 8 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{3} + x^{2} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + x + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} + x^{2} + x}\right ) \]
-sqrt(3)*arctan((sqrt(3)*x^2 + 2*sqrt(3)*(x^4 + x^2)^(1/3)*x + 4*sqrt(3)*( x^4 + x^2)^(2/3))/(8*x^3 - x^2 + 8*x)) - 1/2*log((x^3 + x^2 + 3*(x^4 + x^2 )^(1/3)*x + x + 3*(x^4 + x^2)^(2/3))/(x^3 + x^2 + x))
\[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x^{2} + x + 1\right )}\, dx \]
\[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
\[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
Timed out. \[ \int \frac {-1+x^2}{\left (1+x+x^2\right ) \sqrt [3]{x^2+x^4}} \, dx=\int \frac {x^2-1}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]